On the Spread of Certain Normal Matrices

ON THE SPREAD OF CERTAIN NORMAL MATRICES

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES AND RESEARCH

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

MATHEMATICS

UNIVERSITY OF REGINA

Yongjun Xing

Regina, Saskatchewan

June 2011

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FACULTY OF GRADUATE STUDIES AND RESEARCH

SUPERVISORY AND EXAMINING COMMITTEE

Yongjun Xing, candidate for the degree of Doctor of Philosophy in Mathematics, has presented a thesis titled, On the Spread of Certain Normal Matrices, in an oral examination held on June 8, 2011. The following committee members have found the thesis acceptable in form and content, and that the candidate demonstrated satisfactory knowledge of the subject material.

External Examiner: "Dr. Fuzhen Zhang, Nova Southeastern University

Supervisor: Dr. Shaun Fallat, Department of Mathematics and Statistics

Committee Member: "Dr. Douglas Farenick, Department of Mathematics and Statistics

Committee Member: Dr. Chun-Hua Guo, Department of Mathematics and Statistics

Committee Member: Dr. Boting Yang, Department of Computer Science

Chair of Defense: Dr. Howard Hamilton, Department of Computer Science

*Not present at defense Abstract

A spread of a matrix has extensive and practical applications in some combina torial optimization problems and cybernetics problems. The spread of a matrix is simply defined as the maximum absolute value of the difference between any two eigenvalues of that matrix. There are many existing papers dealing with bounding the spread of a matrix in general. Of interest to us is the spread ofnxn normal matrices with entries in a closed set. In this thesis, we are interested in the classes of real symmetric matrices, real skew-symmetric matrices, complex Hermitian matrices and complex skew-Hermitian matrices, and we determine the structure of these ma trices, in each class, when their spread attains a maximum value. Matlab is used as a tool to aid the verification of some cases. Motivated by some existing results about eigenvalue inclusion regions via discs, we try to build a connection among the disc coverings, spread, and principal submatrices.

i Acknowledgements

I would like to acknowledge the strong and helpful support of my supervisor Dr.

Shaun M. Fallat. Without his help and patience, this work would never have been brought to a conclusion. I would like to acknowledge the generous financial support of the Department of Mathematics and Statistics at the University of Regina, and of the

Faculty of Graduate Studies and Research at the University of Regina. I would like to acknowledge the careful reading and the helpful suggestions from my committee.

ii Contents

Abstract i

Acknowledgements ii

Table of Contents iii

List of Figures vi

1 Introduction 1

1.1 The Spread of a Matrix (History and Motivation) 1

1.2 Notation and Terminology 3

1.3 Outline of the Thesis 6

2 Preliminary Results 8

2.1 Known Results 8

2.1.1 Background 8

2.1.2 Lower Bounds for the Spread of a Normal Matrix 9

2.1.3 Upper Bounds for the Spread of a Graph 14

2.2 Consequences, Facts and Other Results 25 iii 2.2.1 Mirsky's Theorem 25

3 Maximum Spread for Certain Clcisses of Normal Matrices 29

3.1 Case of Real Normal Matrices 29

3.2 Real Symmetric Matrices 35

3.2.1 Structure of Real Symmetric Matrices Whose Spreads Attain

the Bound from Mirsky's Theorem 35

3.3 Spread of Some Real Symmetric Matrices 54

3.3.1 The Form of Real Symmetric Matrices of Rank 2, with Entries

a or 1 54

3.3.2 The Spread of Real Symmetric Rank 2 Matrices with Entries a

or 1 61

3.4 Conclusion about Real Symmetric Matrices 74

3.4.1 Maximum Spread of Real Symmetric Rank 2 Matrices with

Entries in the Interval [a, 1] ( — 1 < a < 1 ) 74

3.4.2 Maximum Spread of Real Symmetric Matrices with Entries in

the Interval [a, b] 79

3.5 Real Skew Symmetric Matrices 81

3.5.1 Structure of Real Skew Symmetric Matrices Whose Spreads

Attain the Bound in the Mirsky's Theorem 85

3.6 Case of Complex Normal Matrices 97

iv 3.6.1 Structure of Complex Hermitian Matrices Whose Spreads At

tain the Bound in Mirsky's Theorem 97

3.6.2 Complex Hermitian Matrices with Entries of Modulus 1 . . . . 102

3.6.3 Skew Hermitian Matrices with Entries of Modulus No More

than 1 104

4 Spectral Inclusions for Normal Matrices 106

4.1 New Spectral Inclusions for Normal Matrices 106

4.2 Imbedding Conditions for Normal Matrices 112

4.3 The Spread of a Normal Matrix and Its Principal Submatrices .... 116

5 Conclusion 119

5.1 Summary 119

5.2 Future Considerations 121

6 Appendix 123

6.1 The Matlab Programs 123

References 128

v List of Figures

3.1 Two cases of the combination parameters in real skew symmetric matrices 86

3.2 Case for both arguments in quadrant I 86

3.3 Subcases 1 and 2 87

vi Chapter 1

Introduction

The central theme in this dissertation is to investigate the maximum spread of cer tain normal matrices. We will introduce necessary notation and preliminary material followed by an outline of this dissertation. In addition, we will give brief introductions within each of the chapters.

1.1 The Spread of a Matrix (History and Motivation)

The spread of a matrix has been of research interest for many mathematicians since the 1950's. There is considerable literature on the spread of an arbitrary matrix, see, e.g., Mirsky (1956); Johnson et al.(1985); Thompson (1992); Nylen and Tam (1994);

Zhan (2006). This quantity has applications in combinatorial optimization problems

[8]. Recent research has shown a connection to graph theory. In Petrovic (1983),

Petrovic determined all minimal graphs whose spreads do not exceed 4. In Gregory et al. (2001), Gregory, Hershkowitz, and Kirkland presented some lower and upper bounds for the spread of a graph, and show that the unique graph with minimum 1 spread among all connected graphs of a given order is the path. However the graph (s) with maximum spread is still unclear, but some conjectures on this topic appear in the paper [9]. In Nikiforov (2006), Nikiforov considers properties of linear combinations of some extreme eigenvalues of a graph. He presents a theorem involving the limit of certain combinations as the order of a graph goes to infinity, and gives an upper bound for the sum of the first two largest eigenvalue of all graphs of fixed order. In Li et al. (2007), Li, Zhang, and Zhou determine the unique graph with maximum spread among all unicyclic graphs on at least 18 vertices. Such a graph is obtained from a star by adding an edge between two pendant vertices. In Fan et al. (2008), Fan,

Xu, Wang, and Liang show that the star is the unique tree with maximum Laplacian spread among all trees of given order, and the path is the unique one with minimal

Laplacian spread among all trees of given order.

Generally speaking, the spread of a square matrix A with order n is defined as

SP(A) = max |Aj — A,|, i,j where Aj, Aj are eigenvalues of A, 1 < i < j < n. When all the eigenvalues are real,

SP(A) = Aj — A„, where the eigenvalues of A are ordered as Aj > A2 > • • • > An.

The study of normal matrices is another research field for mathematicians because this class of matrices has a series of beautiful properties, such as unitary diagonaliza- tion. Some papers investigate the spread of a normal matrix, like [26, 27]. However, there is little research involving the spread of a normal matrix whose entries are re stricted in some range. Our main purpose is to investigate the maximum spread of a certain normal matrices whose entries are in an interval or a circle, and determine 2 the structure of a corresponding extreme normal matrix.

1.2 Notation and Terminology

As mentioned previously, let A € Cnxn, the spread of A is defined as SP(A) = maxjj |Aj — Aj|, where A», \j are eigenvalues of A, 1 < i < j < n; when all the eigenvalues are real, SP(A) — Ax — An, assuming that the eigenvalues of A are ordered

n n as Aj > A2 > • • • > An. Denote the set of eigenvalues of A € C * as cr(A)\ and denote

A(y4), set of singular values of matrix A, which are the square roots of the eigenvalues of the nonnegative Hermitian matrix AA*, where A* is the conjugate transpose of A.

A complex n x n matrix A is a normal matrix if AA* = A*A, i.e, a normal matrix commutes with its conjugate transpose. If A is a real normal matrix, then

ATA = AAT, where AT is the transpose of A.

nxn Denote tr(A) as the trace of the matrix A, i.e. tr(A) = JZ"=1 A = [a»j] € C , and denote ||^4||f as the Frobenius norm of A, i.e.

\\A\\% = tr(A-A) = K'l2- l

n T T Let x, y € C , x = (xi,X2, • • •,x n) , y — (yi,y2, • • •, yn) - Then denote the inner product of vectors (x,y) = y*x = yiXt.

Let

W(A) = {(x, Ax) : (x, x) = 1} denote the numerical range of matrix A. When A is a normal matrix, W(A) is the convex hull of the spectrum of A, denoted co(a(A)). 3 Let A = [ay] € Cnxn, B = [&„] € Cnxn, denote C = A o B = [c*j] €

Cij = a.ij x bij. This product is called the Schur (or Hadamard) product of A and B.

A simple graph is an undirected graph that has no loops and no more than one edge between any two different vertices. A complete graph is a simple graph in which every pair of distinct vertices is connected by a unique edge. A bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint sets V\ and V2 such that every edge connects a vertex in V\ to one in V2\ that is, V\ and V2 are independent sets. A complete bipartite graph, G := (Vi |J V2,E), where 14, V2 are the sets of vertices and E the set of edges of G, is a bipartite graph such that for any two vertices, Vi € Vi and v2 € V2, ViV2 is an edge in G. The complete bipartite graph with partitions of size |Vi| = a and \V2\ = 6, is denoted by Ka^.

The degree of a vertex Vi of a graph is the number of edges incident to the vertex, and is denoted by dj. Let G be a simple graph of order n with degree sequence dud2,..., dn. Define Mx = ^=1 <%•

The Laplacian matrix of a graph G is defined as L = L(G) = D(G) — A(G), where

D{G) = diag{di,..., dn} is a diagonal matrix with the degree of a vertex Vi of G as the diagonal entries, A(G) is the adjacency n x n matrix of G given by: = 1 if vertices vt, Vj are adjacent and ai:> = 0 otherwise. The Laplacian spread of the graph

G is defined as SP(G) = Ai(G) — An_i(G), since the eigenvalues of L can be arranged as follows: Ai(G) > A2(G) > • • • > An(G) = 0, where Ai(G) is the spectral radius of

L, and An_i(G) is called algebraic connectivity.

For a real number a, let a+ equal a if a > 0 and 0 otherwise.

4 For a, (3, 7 6 F", if there exist u, v € F such that 7 = ua + u/?, then we call this pair, (u, v), the combination parameters of a, j3.

Denote round(x) an integer number closest to x. Examples for clarity: round( 1.5) =

1 or 2; round(2.2) = 2.

Denote rank(A) the rank of a matrix A

Let Jn denote a matrix of order n, whose entries are all ones, and enxi =

T [1, ..., 1] € M", let Nn[ a, 6] denote the set of n x n normal matrices all of whose entries lie in the interval [a, b], and let 5n[ a, b] denote the set ofnxn real symmetric matrices, contained in iVn[a, 6]. Let denote the set of unitary matrices, i.e, for any matrix U € &n, UU* = U*U = I. For an n x n real symmetric matrix A, we always denote the eigenvalues of A in decreasing order, Ai(yl) > • • • > An(A).

Denote by S^-fa, 6} as the set ofnxn real symmetric matrices whose entries are either a or b.

Denote by 55„[—a,a] the set ofnxn real skew symmetric matrices with entries in the interval [—a, a].

Let be the class of n x n Hermitian matrices and the class of n x n skew Hermitian matrices.

Two matrices A, B, with the same order, are said to be D-similar if there is a diagonal matrix D with diagonal entries equal to 1 or -1 such that DAD = B.

In Chapter 2, we define a special class of matrices, called Mirsky matrices. A

Mirsky matrix A is a matrix which attains the bound in (2.25) and therefore has the property that n — 2 eigenvalues coincide and are all equal to the arithmetic mean of

5 the remaining two eigenvalues of A, and A is normal.

In Chapter 4, we denote by Ai the (n — l)x(n — 1) principal submatrix of A obtained by deleting the i-th row and column from A.

Let A = [dij] € Nn. Define

.1/2 Vi = and let D(a,r) be the closed disk with radius r € K+ and center a € C.

For real 8, define

H = {a+ ib : a> 0, or a = 0 and b > 0}.

In this case, the lexicographic order is <0, and we have

1.3 Outline of the Thesis

This dissertation is organized into five chapters: In Chapter 1 (this chapter), we present some background and notation. In Chapter 2, we introduce some results on the upper bounds and lower bounds of spread for a general matrix and a graph and then we survey preliminary results and related facts. In Chapter 3, we show the main contribution: maximum spread of normal matrices with entries in an interval or a circle and discuss the related structure of the extreme matrices. In Chapter 4, we review Borcea's conjectures on the eigenvalues of a matrix being covered by the circles of the related principal submatrices and we give some conjectures about the 6 rough upper bound of the spread for the matrices with entries in some range, and then we survey the relation between a normal matrix and its principal submatrices.

In Chapter 5, we summarize the main results in this thesis and discuss some possible future work in this area. Finally, an appendix is included containing the algorithms for finding a Mirsky matrix and a normal matrix with entries in some given range when spread is maximum.

7 Chapter 2

Preliminary Results

2.1 Known Results

In this chapter, we review some results on lower and upper bounds of the spread for general normal matrices and some simple graphs, and we present a key theorem, called Mirsky's Theorem, which inspires our investigation of the maximum spread of a normal matrix with entries in a fixed range.

2.1.1 Background

In the thesis, we focus on the class of normal matrices. Recall that a complex square matrix A is a normal matrix if

AA* = A* A, where A* is the conjugate transpose of A. That is, a matrix is normal if it commutes with its conjugate transpose. If .A is a real matrix, then A* = AT\ it is normal if

ATA = AAT. 8 2.1.2 Lower Bounds for the Spread of a Normal Matrix

In this section, we introduce some classic lower bounds for the spread of a normal matrix.

Recall that of x, y € Cn, then (x, y) denotes the inner product of x and y.

Theorem 2.1.1 ([20]) If A is normal, then

SP(A) = sup\(u,Au) — (v,Av)\ U,V

> a/3 sup |(u, Av)\. (2.1)

If, in addition A is Hermitian, then

SP(A) = 2 sup |(u, Av)\, (2.2) tx,v where the upper bounds above are taken with respect to all orthonormal vectors u,v.

If we let

W{A) = {(re, Ax) : (x,x) = 1} denote the numerical range of A, then it is well known that for normal matrices, W(A) is the convex hull of the spectrum of A, denoted CO(

sup |(u, Av)| = min ||^4 — a/||«p, (2.3) M=IMI=i a (u,v)=0 where || • ||sp denotes spectral norm.

In particular, there are other lower bounds known, and some are presented below.

For example,

SP(A) > 1 _ (2.4) n — 1 i^j and, if Ri denotes the ith row sum, u denotes the standard deviation of the row sums, and .Rjj > Rl2 > • • • > Rin, then (for A symmetric)

f(n+l)/2l 1/2 j f(»+D/2l SP(A)>2v>\l Y, Wj-ft-w)3 >: E(ft,(2-5) n j=1 j=I where [•] denotes the greatest integer part.

Mirsky [19, 20] presented the following lower bounds for SP(A), when A is normal:

SP(A) > \/3max |ai,-|,

SP(A) > max{(Reaa — Reajj)2 + |ai, + a™!2}1^2,

2 2 1/2 SP{A) > max{|ai< - aj:j\ + fla^l - |aji|) } ,

SP(A) > max(|ay| + |a^|), »5*3

SP(A) > mlax^Cij)1'2, (2.6) *¥=j £

2 2 2 2 where = \au - ajj\ + |(a,» - a^) + 4ayajj| + 21a,ij7-1 + 2|a^i| .

10 If A is Hermitian, then:

SP(A) > 2max|aij|,

SP(A) > max{(aji — a,j)2 + 4|aij|2}1/2. (2.7)

Brauer and Mewborn [2] presented the following lower bounds for SP(A) for A normal. If n > 2, let Sj be the trace of any principal submatrix B of order k > 3, and 52 the sum of the principal minors of order 2 of B. Then

|(A; — l)s2 — 2ks2\l^2i k even, SP{A) > { (2.8) 1/2 fc 1/2 (prr) l( - ~ 2fcs2| , k odd.

Let C\ be the sum of all the eigenvalues of A and c2 be the sum of the product of any two different eigenvalues of A respectively. Set K2 = {2(1 — 1 /n)cf — If

A has real roots, then

\ zK , n even, V" 2 (2.9) { (ife)"2**, "Odd. With A Hermitian,

SP(A) > ^ max jo** + ajj + [(a« - a^)2 + 4|ay|2]1/2|

2 2 1/2 - \ min + aSj - [(aii - a#) -I- 4|aij[ ] } , (2.10) and if, in addition, n > 3, Si is the trace of any principal submatrix B of order k > 3, and S2 is the sum of the principal minors of order 2 of B, then

1/2 |{(A; — l)s? — 2A:s2} , k even, SP(A) >{ " (2.11) 1/2 1/2 (prrf) {(& - l)s? - 2ks2} , k odd. 11 Wolkowicz and Styan [31] presented lower bounds for matrices A with real engen- values. Let _2 tr(A2) / £r(.<4)\ n \ n J ' then 25, n even, SP(A) > { (2.12) 2sn/{n2 — l)1/2, n odd. Lower bounds for general matrices axe given in [32].

Considering other choices for the orthonormal vectors u and v in (2.1) and (2.2),

Johnson et al. presented the following lower bound for the spread of a real and normal matrix A.

Theorem 2.1.2 ([16]) Suppose that A is real and normal (or more generally W(A) — co(a(A)). Then 1 SP(A) > n — 1

th th Proof. Choose u = ^enxl and v = ^(e^ — ej), where ek and ej are the k and I standard basis vectors, respectively. Then (2.1) implies that

Ofcfc + o-u — 2aw

" n ? a* ~ 2n(n — 1) + Ull) + n(n-l) ^

Ukl 2"(«-l)?2(n 1)Gfcfc+ n(n-l)^

(2.13) n _ 15Z "a-

12 The result now follows by noting that SP(—A) = SP(A).

13 2.1.3 Upper Bounds for the Spread of a Graph

First we introduce some results about upper bounds for the spread, SP(G), of a graph. The matrix A here is related to the graph and is real and symmetric, and is known as the adjacency matrix of a simple graph G with n vertices. So

SP(G) = SP{A) = Aj — An, if we assume the eigenvalues of A are Ai > A2 > • • • > An.

The following theorem on SP(G) is due to Gregory et al. [9].

Theorem 2.1.3 For a graph G with n vertices and. e edges,

If G has no isolated vertices, then equality holds throughout if and only if equality holds in the first inequality; equivalently, if and only if G = Kaf, for some a, b with e = ab and a + b < n.

Remark 1: A complete graph is a simple graph in which every pair of distinct vertices is connected by a unique edge. Geometrically K3 forms the edge set of a triangle, K4 a tetrahedron, etc.

Remark 2: A complete bipartite graph, G := (Vi + V2, E), where Vi, V2 are the sets of vertices and E the set of edges of G, is a bipartite graph such that for any two vertices, v\ € V\ and V2 £ V2, V\v

Gregory, Hershkowitz, and Kirkland give other results for the maximum spread of a graph with e edges and a graph with n vertices. They consider a complete p-partite graph (precisely p non-empty parts), which is a simple graph (a simple graph is an 14 undirected graph that has no loops and no more than one edge between any two different vertices).

For the first case, maximum spread of a graph with e edges, they have the follow ing,

Theorem 2.1.4 ([9]) Let G be a graph with n vertices, e edges, and precisely k negative eigenvalues, 1 < k < n — 1. Then

(2.14)

Equality holds if and only ifG has at most three distinct non-zero eigenvalues: \\, /3, A„, where \i > 0 > (3 > \n and (5 has multiplicity k — 1 if /3 > \n and multiplicity k if

(3 = A„. Equivalently, equality holds if and only if GQ is a complete (k + I)-partite graph and, when k + 1 >4, the k smallest parts of Go all have equal size (necessarily,

= (Ai + A„)/(/c — 1)).

They prove the above theorem for the values of k = 1,2 and k > 3 using the characteristic polynomial of the adjacency matrix of G. We can see that the partial derivative with respect to k of the upper bound in the theorem above is non-negative for all real k > 2. Consequently, the inequality in the theorem above holds if G has at least 2 and at most k negative eigenvalues. And the partial derivative with respect to Ai of the upper bound in the theorem above is

where A is the square root in (2.14). It implies that the upper bound in Theorem

2.1.4 is strictly increasing in Ai when Ai < y/e and strictly decreasing when Ai > sfe. 15 They then present the following corollary.

Corollary 2.1.5 ([9]) Let G be a graph with n vertices, e > [ri2/4j edges, and at most k negative eigenvalues. Then (I)'

Equality holds if and only if (k + l)|n and G is a regular complete (k + I)-partite graph.

Since e > [n2/4J, Ai > ^ > y/e, we can derive the inequality above with replacing

\\ by Equality holds if and only if equality holds in Theorem 2.1.4 and \\ = which means that G is a regular and a complete (k + l)-partite graph.

For the second case, SP(n), the maximum possible spread of a graph with n vertices, they have some upper bounds.

By studying the relation between H, an induced subgraph of G on n — 1 vertices and G, a connected graph on n vertices, they find the next lemma.

Lemma 2.1.6 ([9]) If H is an induced subgraph of G, then Xn(G) < An(i/) <

\i(H) < Ai(G). Thus SP(G) > SP{H) with strict inequality if G is connected and H is a proper induced subgraph of G.

It follows that SP(n) is strictly increasing, and they present a conjecture about the maximum spread SP(n).

Conjecture 2.1.7 ([9]) The maximum spread SP(n) of the graphs of order n is attained only by G(n, [2n/3j); that is, SP(n) = |_(4/3)(n2 — n + 1)J1//2 and so

(l/v/3)(2n - 1) < SP(n) < (l/v/3)(2n - 1) + y/3/(4n - 2). 16 Then they verify that G = G(n, |_2n/3j) if G is one of the these following graphs on n vertices and G has maximum spread, according to the value of k.

If A: = 1, then G is a complete bipartite and so, by Theorem 2.1.3, SP{G) —

2y/e 3, and G — K2 = G(n, |2n/3j) when n — 2.

If k = 2, then G is a (complete) tripartite graph and Proposition 2.1.9 (see below) implies that SP(G) < SP(n) when n > 35.

If k > 3, the extremal graphs G in Theorem 2.1.4 are the complete (k + l)-partite graphs with k parts of size n\ and a single part of size n2 > n\. Denote such a graph by G(n,k,rii). For this case, they first show that n\ = 1. Since SP(n) is strictly increasing, only graphs without isolated vertices need to be examined. Thus, n = krii + n2. It follows from the discussion in the proof of Theorem 2.1.4 that the extreme eigenvalues Aj, A„ of G(n, k, n*) are the zeros of the middle factor of the characteristic polynomial of (the adjacency matrix of) G [4, p. 74] given by

or, equivalently, they are the roots of the quadratic equation X2 — ni(k — l)X—kniTi2 =

0. Thus, G(n,k,ni) has spread

Ai — A„ = (•n\(k — l)2 + 4kniri2)1/2 • (2.15)

Since kni = n — n-i and ni(k — 1) < n — — 1 with equality if and only if ni = 1, it follows that, for k > 3, the spread is largest when n\ = 1, as required. When rzi = 1, G(n,k, 1) = G(n, k). Taking ni =1 in (2.15) gives SP(G(n,k)) = ((k — l)2 +

Ak(n — A;))1/2. This quadratic is maximum when k is an integer closest to (2n — l)/3, 17 that is, when k = |2n/3j, as claimed at the outset.

Gregory et al. [9] present some necessary conditions that a graph G on n vertices must satisfy when it attains the maximum spread SP(n).

Lemma 2.1.8 ([9]) Let G be a graph with e edges and let a > 1. If SP{G) > an, then e must satisfy the quadratic inequality 8e2 — (4a + 2)n2e + a2n2 < 0.

Proposition 2.1.9 ([9]) If SP(G) = SP(n) and n > 35, then G is not tripartite.

Proposition 2.1.10 ([9]) If SP(G) = SP(n), then G must be connected.

They then describe some characteristics about the pair of eigenvectors for the extreme eigenvalues when the maximum spread is attained. For a simple graph G of order n, the adjacency matrix A is symmetric. So

T T Ai > x Ax and An < y Ay,

where Ai > A2 > • • • > An, Aj € cr(A),i = 1 and unit vectors x, y € R".

Equalities hold if and only if x, y are the unit eigenvectors associated with Aj,An respectively. Thus

(2.16) where the maximum is taken over all pairs of unit vectors in Rn and is attained only for orthonormal eigenvectors of A corresponding to the eigenvalues Ai,An, respec tively. The entries of x may always be assumed to be non-negative (and positive if G is connected). Define such an ordered pair of orthonormal eigenvectors x,y of

G with x > 0 (i.e, x entry-wise nonnegative) an extremal pair of eigenvectors of G. 18 The following three lemmas are an immediate consequence of (2.16) and Proposition

2.1.10.

Lemma 2.1.11 ([9]) Suppose SP(G) = SP(n) and x, y is an extremal pair of eigen vectors of G. Then distinct vertices i,j of G must be adjacent whenever XiXj —yiUj > 0 and non-adjacent whenever XiXj — yiyj < 0.

Lemma 2.1.12 ([9]) SP(n) = SP((G(x,y)) for some graph G(x,y) with x, y € Rn orthonormal and x positive.

For a real number o, let a+ equal a if a > 0 and 0 otherwise.

Lemma 2.1.13 ([9]) For n > 2, SP(n) = max Ylij(xixj ~ yiVj)+> where the max imum is taken over all pairs x, y of orthonormal vectors in Kn with x entry-wise positive.

As we know, in graph theory, the degree of a vertex of a graph is the number of edges incident to the vertex. The degree of a vertex Vi is denoted di here.

Liu et al. [18] provide an improvement of Gregory's bound on SP(G) for graphs with additional restrictions as followings. Let G be a simple graph of order n with degree sequence rfj, d^, • • •, dn. Define Mi = X)"=i d*. Then the following lemma is a description about Mi. We call G an (n, e) graph if G has n vertices and e edges.

Lemma 2.1.14 ([18]) For any (n,e) graph with f \-cycles,

Mi = ^tr(A4) + e - 4/,

19 where A is an adjacency matrix of G. Furthermore, if Aj, A2,..., An are the eigenval ues of A, then

Mi = if>' + e-4/. <=1

k Proof. Let A = ()and A = (a*•). It is known (see [4]) that af{ is the number of all closed walks of length 4 from vertex i to i in G. Let fi denote the number of

4-cycles located at vertex i. Denote i ~ j as vertex i is adjacent to vertex j. Then

ati = di + -1) +2 -^

= d% + dj — di + 2fi.

Thus,

trM4) = E4

1=1 t=l j^i i=1 i=l

= Mi + Mi — 2e + 8/.

A = r 4 4 Hence Mi = \tr(A ) + e — 4/, as X)"=1 * (^ )> trace of A (see [4]). •

They then focus on graphs without K'4s (called K4-free graphs).

Lemma 2.1.15 ([18]) For a K^-free (n, e) graph G, Mi < e2 + e — 4/. If G is connected and equality holds, then |iV(u) U 7V(i»)| = n holds for any {u, v} 6 E(G).

20 Proof. For any edge {u, i>} € E(G), let fuv denote the number of 4-cycles containing edge {u, v}. Since G is a Ktj-free graph, then d(u) + d(v) < e + 1 — fuv. Thus,

2>(«)+<<(«))< + £1(2.17) ti~V It follows that Mi < e2 + e — 4f.

If equality holds in this lemma, by inequality (2.17), it follows that d(u) + d(v) = e 4- 1 — fuv holds for any {u, t>} e E(G). Let G' denote the subgraph induced by

N(u) U N(v), and e' denote the number of edges of G'. Since G (and then G') is

/Cj-free, then e + 1 — fuv — d(u) + d{v) < e' + 1 — fuv < e + 1 — fuv, implies that e = e'. Moreover, since G is connected, then |N(u) U N(v)| = n. •

The following corollary implies that if Af > 2e2, then G contains at least one K4.

2 Corollary 2.1.16 ([18]) For a K4-free (n, e) graph, ]T"=1 Af < 2e , where equality holds for G = Kaj or G = K3.

Proof. Combining Lemma 2.1.14 and 2.1.15, we have

\ ^2 xt + e ~ 4/ < e2 + e - 4f. <=i

2 Hence, A* < 2e . If G = Ka,b or G = K3, it is easy to check that d(u) + d(v) = e+1 — fuv holds for any u, v 6 E{G). Thus, equality holds in inequality (2.17), implies that equality holds. •

21 Let rank(A) the rank of a matrix A. Now the following is from [18] on the upper bound for the spread of a graph.

Theorem 2.1.17 For an (n, e) graph G with eigenvalues Ai, A2,..., An,

SP(G) < Ax + \J2M1 - 2e + 8/ - A} < 2^/Ml-e + Af.

If G has no isolated vertices, then equality holds throughout if and only if equality holds in the first inequality; equivalently, if and only if G = Kab for some a, b with e — ab and a + b — n.

Proof. By Lemma 2.1.4, X)"=1 Af = 2M\ — 2e + 8/. Thus Af + A* < 2M\ — 2e + 8/ and -{/2Mj - 2e + 8/ - A? < An < $/2Mx - 2e + 8/ -

Therefore,

SP(G) = A, - An < Aa + ^2Mi -2e +8/ - Af.

Equality holds if and only if A2 = A3 = • • • = An_j = 0, that is, if and only if .4(G) = 0 or rank(A) = 2, equivalently, if and only if the non-isolated vertices of G have at most two distinct neighborhood sets. Thus, equality holds if and only if e = 0 or

G = Kayb for some a, b with e = ab and a + b = n. If e = ab, then Ai = — An = y/ab.

Note that Ai + \j2Mi — 2e + 8/ — Af is a strictly increasing function of Ax when

Ai < \JM\ — e + 4/, and it is strictly decreasing when Ai > \/Mi — e + 4/, we have

SP(G) < Aa + \J2M1 -2e + 8f-\j < 2i/M1-e + 4f.

All equalities hold when Ai = — An = Va6. •

22 The bound of Theorem 2.1.17 is better than Gregory's Theorem since for a AT4-free

(n, e) graph, by Lemma 2.1.14 and Corollary 2.1.16, we have 2M\ — 2e + 8/ — A* =

E?=i 2e2 - Xi < 4e2 - 4eA? + Thus

Now we turn to the maximal Laplacian spread of a tree in Fan et al. [6]. The

Laplacian matrix of a graph G is defined as L = L(G) = D{G) — A(G), where

D(G) = diag{di,,d n} is a diagonal matrix with the degree of a vertex Vi of G as the diagonal entries, A(G) is the adjacency n x n matrix of G given by: a,ij = 1 if vertices Vi, Vj are adjacent and = 0 otherwise. It is well known that L is symmetric and positive semidefinite so the corresponding eigenvalues of L can be arranged as follows:

Ai(G) > A2(G0 > • • • > An(G) = 0.

The Laplacian spread of the graph G is then defined as

SP(G) = Xi(G) - A„_j(G),

where Ai(G) is the spectral radius of L, and An_j(G) is called algebraic connectivity, which has a beautiful structural property given by Fiedler [7].

Theorem 2.1.18 ([6]) Forn > 5, the star is the unique tree with maximal Laplacian spread among all trees of order n.

23 Remark: By Theorem 2.1.18, the maximal Laplacian spread among all trees of order n is n — 1, and the related matrix L must then be, \ 71 — 1

\ -e(n-l)xl In-

24: 2.2 Consequences, Facts and Other Results

We will introduce an important result, Theorem 2.2.3, which provide an upper bound on the spread of a matrix, and describes when this bound is attained. Using this fact, we will focus on some certain normal matrices, and find the maximum spread for this class of normal matrices, and determine the corresponding structure of the matrices.

2.2.1 Mirsky's Theorem

Lemma 2.2.1 Let z\,...,zn be arbitrary complex numbers, n>3,t> 1, and denote

s = max |Zi — zA, (2-18)

M = (2.19) 1=1

Then

s < 2(M/2)1/t, (2.20)

with equality if and only if zp + zq = 0 for some p,q E {1,..., n}, p ^ q, and Zk = 0 for k^p,q.

Proof. From (2.19), we know \zj\ < M1^. Assume \Zi\ = rM1^, where r G [0,1].

Then for j ^ i we have

\zj\ < M1/4( 1 - r')1/4 and consequently,

1 1 \zi ~ ZjI < \zt\ + \Zj\ < M^[r + (1 - r ) '*] (t ± j). (2.21) 25 The function: r —• r + (1 — r')1/' has a maximum equal to 21 when rl — It follows that

l t \zi-zj\<2(M/2) / = (2.22) whence the inequality (2.20) holds at once. One can easily see from (2.21) that in order to have equality in (2.22) one must have \z{\ = \zj\ = (M/2)1/' and argzj = ir + argZi.

These relations, together with the definition of M, imply that Zi 4- Zj = 0 and 2* = 0 for k ^ i, j. Clearly, these latter conditions are sufficient in order to have equality in

(2.21) and therefore also in (2.20). •

Let £ €

A — £1 are ordered as |Ai(£)| > |A2(£)| > ••• > |An(^)|; recall that A (A) denotes the set of singular values of A, namely, the square roots of the eigenvalues of the nonnegative Hermitian matrix AA*, and let Aj(£) be the ith element in A(A — £1).

We have the following lemma.

Lemma 2.2.2 Let A be a complex n x n matrix (n > 3), let £ € C, and let t > 1.

Then

SP(A) < 2< ^ <2-23) t=i

Proof. Note that SP(A) = SP(A - £/). By H.Weyl's Theorem [29], we have £>;«)!(«)- (2-24) i=l i=l

Now, the statement of this lemma follows from an application of Lemma 2.2.1. •

26 We axe now in a position to state and prove a very important result, due to Mirsky, on the spread of a general matrix. The proof technique is similar to that of Mirsky [19].

Recall that tr(A) is the trace of the matrix A, tr(A) = an, A = [a^] € CnXT\ and that ||v4||f denotes the Frobenious Norm of A, i.e. ||^4||f = tr(AA*), and when

2 A is normal, \\A\\ F = E

Theorem 2.2.3 (Mirsky[19]) Let A be a complex n x n matrix (n > 3.) Then

2 SP(A) < J2\\A\\F --\tr(A)\\ (2.25) V n with equality if and only if A is normal and n — 2 of its eigenvalues are equal to each other and this common value is the arithmetic mean of the remaining two.

Proof. We have

i~l = tr{A*A -£A -£A* + |£|2/)

= Mllf" jWA)|J + n|{-iir(A)|2 («6C).

Clearly, the choice £ = yields the smallest possible value for this last ex pression. Taking, in (2.23), t = 2 and £ = we obtain the inequality (2.25). It is known that, in the case t = 2, we have equality in (2.24) if and only if A — £/ is normal, i.e., if and only if A is normal. This, together with the conditions for equality in (2.20), applied to the eigenvalues of A — £1, gives the conditions for equality in

(2.25). •

27 Remark: Here we write the condition, if n complex numbers axe such that n — 2 among them are equal to each other and this common value is the arithmetic mean of the remaining two, as condition .

When the characteristic roots of A are all real, the following result will yield an upper bound generally better than that of Theorem 2.2.3.

Theorem 2.2.4 (Mirsky[19]) If all characteristic roots of the n x n matrix A are real, then

where t^A is the sum of all 2x2 principal minors of A, with equality if and only if the characteristic roots satisfy ¥>.

Note: The reason why Mirsky's result is useful for this work is because: when the spread of A attains the bound in (2.25), then A is a normal matrix and has a very special property of the eigenvalues. Throughout my thesis, we want to find the maximum spread of an n x n normal matrix, in some sense, which has a connection to the condition ^ and the rank of A.

28 Chapter 3

Maximum Spread for Certain Classes of

Normal Matrices

In this chapter, we focus on the maximum spread and the related structure of certain classes of normal matrices, including real symmetric matrices with entries in a specific interval, real skew symmetric matrices, complex Hermitian matrices with entries of modulus no more than 1, and skew Hermitian matrices with entries of modulus no more than 1. In this discussion, Theorem 2.2.3 (Mirsky's Theorem) plays an important role, as outlined in the note in the previous section.

3.1 Case of Real Normal Matrices

Let Jn denote a matrix of order n, whose entries are all ones, let Nn[ a, b ] denote the set of n x n normal matrices all of whose entries lie in the interval [a, 6], and let

Sn[ a, 6] denote the set ofnxn real symmetric matrices, contained in JV„[ a, b]. Let

denote the set of unitary matrices, i.e, for any matrix U € UU* = U*U = I.

29 For an n x n real symmetric matrix A, we always denote the eigenvalues of A in decreasing order, Ai(>l) > • • • > An(v4).

Recall from Chapter 2 that the spread of an n x n matrix A is defined as

SP{A) = max |Af(yl) — Aj(A)|.

This quantity has applications in certain combinatorial optimization problems [8]. In this chapter we investigate the maximum value of SP(A) for A € Nn[ a, 6].

Lemma 3.1.1 ([11]) Let A € M„(R), then A is normal if and only if there exists a real orthogonal matrix Q € Mn(R), such that

/ \ A\

QTAQ = 0 & Mn{R), 1 < k < n, 0

Ak where Ai is either a real 1-by-l matrix or is a real 2-by-2 matrix of the form, ( \ A = \ -bi ^

T th We also write Q = (qi, q2, • • •,q n), where qi = (qi^q^, • •• ,qin) is the i column of Q.

Corollary 3.1.2 A € M„(R) is a normal matrix. Then A is symmetric if and only if cr(A) C R.

30 Lemma 3.1.3 Suppose A € Mn(R) is a normal matrix. If SP(A) attains the upper bound from Theorem 2.2.3, then either all the eigenvalues are real or there is exactly one pair of complex conjugate eigenvaleues. In the latter case, the remaining n — 2 real eigenvalues are equal to the real part of the complex eigenvalues.

2 2 Proof. By Theorem 2.2.3, if SP(A) = \J2\\A\\F — ^\tr(A)\ , then A is a normal matrix and n—2 of its eigenvalues are all equal to the arithmetic mean of the remaining two. Since SP(A) = max^^.gjj^) |A; — Aj|, we consider different cases for calculating the value of this spread.

Case 1: a{A) C R. Then SP(A) = Ai - An and all the other eigenvalues are equal

to . By Lemma 3.1.1, A is a real symmetric matrix and the rank of A is

either 0,2, n — 1, or n.

Case 2: Suppose A has complex eigenvalues, which must occur in conjugate pairs.

Assume 0 < SP(A) = |Aj — Aj| for some pair of eigenvalues A*, Aj.

Subcase 2.1: If Xi: A j € R, then a(A) C R and Case 1 applies.

Subcase 2.2: If Af € R, and Xj = a + ib € C (b ^ 0), then the other eigen

values are equal to + i\ which forces| = —b, and then we reach a

contradiction.

Subcase 2.3: If A» = c+ ib = Xj, then the remaining eigenvalues are equal to

c 6 R. In this case, the rank of A is either 2 or n.

Subcase 2.4: If Af = C\ + ib\, Xj — oi 4- ibi 6 C, but not conjugate, then the

31 other eigenvalues are all equal to , and since conjugate eigen

values must occur in pairs, we reach a contradiction. This completes the

proof. •

Remark (*): If matrix A satisfies the conditions in Lemma 3.1.3, then the possible ranks of A must be one of the following: if SP(A) = |A, — Aj| for some i,j:

• Ai, Aj 7^ 0, Aj + Aj ^ 0 => rank(A) — n

• Aj = 0, Aj 7^ 0, Aj + Aj 7^ 0 or Aj 7^ 0, A j = 0, Aj + Aj ^ 0 => rank(A) = n — 1

• Aj, Aj 7^ 0, Aj + Aj = 0 => rank(A) = 2

• Aj = 0, A j = 0, Aj + Aj = 0 => rank(A) = 0

32 Remarks:

1. In Case 1 above, if SP(A) is maximum and A € S„[a, 1], then, as A is symmetric,

it follows that the entries of A are either a or 1 (see Lemma 3.2.1).

2. In Case 3, let / 1 \ 75 72 $2x(n-2) U = Q —i 72 72

0(n--2)x2 I(n-2)x{n-2)

ai±m 91 -m a„ - ( V2 ' V2 ' i • • • j 9n ^ Then c + ib 0 $2x(n-2) U*AU 0 c — ib 1

0( n-2)x2 1 C-^(n—2)x(n- 1 and c b £ ? > c X 1 QTAQ —b c

0(n-2)x2 C/(n-2)x(n-2)

Now B = A — cl = biqiqj — 92 is a real skew symmetric matrix, which is rank

2, where 91,92 are the associated eigenvectors, qjqi = qjq2 = 1, and qjq2 = qjqi = 0-

Let C = [cfj]nxn = qxqj — 929^. Then ||C||^ = 2 and tr(C) = 0 and 0 < |Cy| =

|(9i)i(92)j — (92)1(91)7! < 1, for z, j = 1,..., n. So B = A — cl = bC, the entries of B are bounded in [—6, b], b > 0, where b is the imaginary part of the complex eigenvalues of A, and c is the real part of those complex eigenvalues of A. Then the diagonal

33 entries of A are all c and the off-diagonal entries of A lie in [-b,6].

We study these two cases, Case 1 and Subcase 2.3, in the next two sections.

34 3.2 Real Symmetric Matrices

In this section, we focus on the class of real symmetric matrices, 5„[a, 6]. We study the maximum spread of these real symmetric matrices and their related matrix structure.

3.2.1 Structure of Real Symmetric Matrices Whose Spreads

Attain the Bound from Mirsky's Theorem

Lemma 3.2.1 Suppose A € Sn[a, 6]. If SP(A) is maximum, then each entry of A is either a or b.

Proof. Throughout this proof we write Aj for Aj(yl). For x € Rn, we always write its components as X\,..., xn. Given A £ 5n[a,6], let x,y € R" be unit eigenvectors such

T T that Ai = x Ax, An = y Ay. Then

SP(A) — XTAX — yTAy = eT[A o (xxT — yyT)\e, e = (1,..., 1)T. (3.26)

Note that the (i,j) entry of xxT — yyT is XiXj — yiyj. We define a new matrix A = (a^) as follows: a.ij = b if Xjxj — yiyj > 0 and 5^ = a if XiXj — yiyj < 0. Then from (3.26) we have

SP(A) < xTAx — yTAy

< max{u>TAu> : ||a;|| = l,a; G Rn} — min{zTAz : ||z|| = l,z E K"}

= SP(A).

35 We will show that when SP(A) is maximum there is no position (i,j) in the re lated matrix such that XiXj — yij/j = 0 (see Theorem 3.4.2 below). Therefore the maximum spread for a matrix in .?„[ a, b ] can only be attained at some {a, 6}-matrix

(that is, a matrix in 5n[ a, £>] whose entries are either equal to a or b). •

As a consequence of Lemma 3.2.1, we need to determine the structure for a matrix

2 2 A in Sn-{a, b} (symmetric nxn matrices whose entries are either a or b) when a < b , since SP(A) = SP(—A). Furthermore by considering b~1A instead of A, we only study the maximum spread for matrices in Sn-{a,1} ( — 1 < a < 1).

We now focus on rank 2 matrices A E 5„-{a, 1} ( — 1 < a < 1 ). In this case, we can find an invertible principal submatrix of order 2. Assume, by simultaneous permutation of rows and columns of A, that this 2x2 submatrix lies in rows/columns

1,2. Thus every other row in this matrix is a linear combination of these two rows.

From which we determine the structure of these matrices. Examples for n = 2 and

36 n = 3 are given below (each with all real eigenvalues):

/ \ I \ a 1 1 a 1 Al = ^3 1 1 1 1 1 a 1 1 1 1 / \ \ a 1 a a a a

*§ = 1 1 1 a I a

^ Q 1. Oh j a a a ( \ a a a i \

^3 a 1 1

1'/ \ /

We will show that SPIA^) is maximum when n = 2 and SP(A\) is maximum when n = 3.

From Chapter 2, we know that a symmetric real matrix order n (n > 3), named a

Mirsky matrix, will attain the largest possible spread (see Theorem 2.2.3) when n — 2 eigenvalues coincide and are all equal to the arithmetic mean of the remaining two eigenvalues.

We now focus on the structure of a Mirsky matrix A, which lies in the class

5„-{a, 1}, with a € [—1, 1). Take

_ tr(A) _ Ai(A) + An(^4) * ~ n ~ 2

37 and let

A' = A- £/.

Then, using Remark (*), we have

SP{A') = SP(A), tr(A') = 0, rank(A') = 2.

We call A' the transform of A. In the following, we explore the structure of A'.

Suppose that there are s a's and (n — s) l's on the main diagonal of A and observe tr(A) = as + n — s, with 0 < s < n. Let

x — —(a — 1 )(n — s) € [—2,0], (3.27) n V = ~(1 — a) G [0,2], (3.28) n y — x = 1 — a > 0. (3.29)

So there are si's, (n — s) y's on the main diagonal of A' while any off-diagonal entry is either a or 1 in matrix A'. Since rank(A') = 2, by simultaneous row-column permutation, we may assume that the upper left 2x2 principal submatrix of A' is nonsingular and therefore must be one of the following 6 cases: / / ( \ X 1 x a J y i

a x * x \ ) K1 y J Case 1 Case 2 Case 3 / N (3.30) y a ( x a\ (x l) a y, 1 a y \ 1 "J \ ) Case 4 Case 5 Case 6 38 Set a, (3 as the first and the second row vector of A' respectively, and let 7* be a remaining arbitrary row vector. Then 7< = U

Remarks:

1. x ^ — 1 in Case 1; |x| ^ |o| in Case 2; y ^ 1 in Case 3; y |a| in Case 4; neither x nor y is 0 in Case 5 and Case 6.

2. When n = 2, there are 4 types of Mirsky matrices A',

( \ F \( \ / \ 0 1 0 a x 1 x a

1 0 \ Q °/ 1 —x a —x / \ / •where x = ^ £ [—1> 0).

Consequently, the general structure of such a Mirsky matrix order n is the follow ing:

mi p aJ\ xs! aj\ X32 JlXS3 <^lX«4

P m2 Q-JlXSl J1X32 0"J\XS3 JlXS4

Q-JsiXl •ElJsi VlJsi XS2 ViJsxxai yz

@"^82 X1 J32XI I/1J32XSI %2

Js3*l Q,Js3Xl ?/2<7s3XSl ViJS3XS2 S3XS3 2/6^3x44

JSAXI J34 X 1 2/3«7S4XSJ ybJs4X$2 ye J34 xS3 -£4 «7S4 X34

39 where si,s2,s3,s4 € [0,n - 2], sx + s2 + s3 + s4 = n - 2, n > 3, m1,m2 € {x,y},

and p € {a, 1}. When s* > 2, Xi € {1, a}; yj G {1, a}, i = 1,... ,4, j = 1,..., 6.

Following is our main observation regarding the possible structure of a Mirsky

matrix.

Theorem 3.2.2 If A is a Mirsky matrix in ^-{a, l},a € [-1, l),n > 3, then A',

the transform of A, must be in one of the following ten forms:

( \ ( \ —IT "S1XS2J —-2 7S 1XS3 2a- 1

—IT —IT J..82* Si 2 s2 2 1 (2a 1) aJix(n—2)

y — -2^ JS3XS1 — -2 "33x32 J '33 Q 0>J(n-2)x\ ^Jn-I j

n 3 1 < si, s2 < n - 2, s3 = |(si + s2) °= ^2' ^

/ / \ aJs Q.Jsx(n-s) Oi/s Jsx(n—s)

^ 0

a = s = 1 : n — 1 a = —s = 1 : n — 1 .9 '

/ \ ( &Jn—l11 l)xl ^ ftJn—1 J(n-l)xl

^ a«/lx(n—1) 2d 1 ^ «/ix(n—1) 2a — 1

a = 1+n a l+n

40 ( \ 0«1 Js\X32 Osi XS3

J32XS1 0«2 0^2 X 53

Os3XSl 0^3X32 where 1 < si, S2 < n — 1, si + + S3 = n,

{ \ a=l 1

1 —3 1A a 1 2^-°) a 1 3 / where a is the unique solution of 11a3 — 15a2 — 57a + 7 = 0, with a £ [—1, 1),

l^a ^ 3 x a

1A 3 1 a 1 2^a'1) a 1 3 / where a is the unique solution of 11a3 — 15a2 + 51a + 7 = 0, with a € [—1, 1),

1—a a a

1—a

a 1 2^11 where a is one of the two solutions of 7a3 — 57a2 — 15a + 11 = 0, with a € [—1, 1).

41 Proof. The order 2 nonsingular principal matrix is in one of the 6 cases showed in

(3.30). We then determine the structure of a Mirsky matrix A' as the other rows must be linear combinations of these first two rows, we include the proof here for Case 1, the remaining analysis may be performed in a similar manner for the other cases.

Let B[,B'2 be the leading order 3 and order 4 submatrices, respectively, that contain the order 2 principal matrix as in Case 1 in (3.30). We will determine the structure of a Mirsky matrix A' in two steps.

Step 1: Study the structure of the Mirsky matrix with submatrix B[, which has the same remaining row vectors starting from the 3rd row, and let i = 1,... ,4, be the indexes of the respective blocks in matrix (3.31) and (u, v) the combination parameters for the remaining rows in A'. We begin with case analysis starting with

Case 1 from (3.30). This analysis will be divided into four subcases as follows:

Case 1.1.1: \ X 1 a u

1 X a V

a a X

It follows that

ux + v = a => (it — v)(x — 1) = 0, u = v since x < 0, so 2ua = x,u(\ + x) = a. u + vx = a 42 If u = 0, then a = 0, x = 0 and s = n; else u ^ 0, then a / 0, a:2 + x — 2a2 = 0. If

Si > 2, then x = a < 0, which implies a2 + a — 2a2 = 0, a contradiction, so si < 1, and when n = 3, s — 3, then x = 0 = a, u = v = 0, a contradiction.

Case 1.1.2: \ X 1 a u

1 X a V

a a yj

By Case 1.1.1, u = v, so 2ua = y, «(1 + x) = a. If u = 0, then y = 0 = a, s = 0, a contradiction; so u ± 0, and it follows that a ^ 0, (1 + x)y = (1 + x)(l +x — a) = 2a2.

Then

„ 1 na 1. x = 2a-l, y — a, u = v = s = a € (0,/n -);

or x — —1 — a, y = —2a, u = v = —1, s = — Jna , a € (—1, 0). 1 — a

If sj = n — 2, it follows that y — a, x = 2a — 1, a = or y = —2a, x = —I — a, and a =

Case 1.2.1: \ X 1 a u

1 X 1 V

a 1 X

43 It follows that

> ux + v = a < => (u — v)(x — 1) = a — 1 and (u + v)(x + 1) = a + 1. u + vx = 1

If a = —1, then u = —v, so au + v = 2v = x, 2v(x — 1) = 2, which means x2 — x — 2 = 0, x = —1, a contradiction. So a ^ —1, and u =

Then we have (x — a)(x2 + ax — 2) = 0 by au + v = x. If S2 = n — 2, then it follows that a: = a = 0.

Case 1.2.2: \ X 1 a u

1 X 1 V

a 1 yj

By Case 1.2.1, a ^ — 1 otherwise u = —u, au + v = 2u = y, 2v(x — 1) = 2, which means y(x — 1) = 2, obviously a contradiction. So we have

x3 + (1 — a)x2 — (a2 + 2)x + (3a — 1) = 0.

2 2 If s2 > 2, then y = 1, x = a < 0, it follows that a — 1 = (a + l)a — 2a, a = — 1, a contradiction; or y = a > 0, x = 2a — 1 <0. Hence 2a3 — 3a2 + 1 = 0, but there is no root in the interval (0, |), so s2 < 1. When n = 3, it follows

s 3 2 x = ^pJ y — |(1 — a), 11a — 15a — 57a + 7 = 0 and there is a unique solu tion a in the interval (—1,1). 44 By symmetry, similar results hold in Case 1.3 as in Case 1.2.

Remark: Case 1.1, Case 1.2, Case 1.3, and Case 1.4 mean the 3rd row vector in A' follows as (a, a,...), (a, 1,...), (1, a,...), and (1,1,...), respectively. Case 1.1.1 and

Case 1.3.2 mean the 3rd row vectors follow as (a,a,x,...), (1, a, y,...). The similar idea is applied to the other cases.

Case 1.4.1: \ x 1 1 u

1 X 1 V

1 1 X

It follows that

ux + v = 1 => (u — v)(x — 1) = 0, u = v since x < 0. u + vx — 1

So 2u = x, 2u{\ + x) = 2, u^O then x2 + x — 2 = 0, x = —2, a contradiction since it implies s = 0. So Case 1.4.1 is impossible.

45 Case 1.4.2: \ X 1 1 u

1 X 1 V

1 1 yj

By Case 1.4.1, u = v, so 2u = y, y(l+x) = 2. If s4 > 2, then it follows that y = a>

0, 1-fx = 2a, 2a2 = 2, a contradiction; or y = 1, x = a < 0, y(l+x) = 1+a = 2, also a contradiction. So S4 < 1, when n = 3, it follows y = |(1—a), x = a2+a+7 = 0, there is no real solution.

Step 2: Now, we study the cases for which there are all possible combinations with the remaining row vectors in a Mirsky matrix A' with submatrix B'2.

If the main diagonal of A' consists of all x's, it follows that x = a = 0. Then it follows that the structure of the Mirsky matrix, after simultaneous row and column permutations, in Case 1 is as follows:

/ \ 0«2+l "^(S2+l)x(«3+l) 0(a2 + l)xst

•^(«3 + l)x(S2+l) 0«3+ l 0(S3+ 1)XS1

^ 0SjX(s2+l) 0siX(s3 + l) 0Sl

where 0 < Sj < n — 2, ^2i s, = n — 2, 1 < i < 3.

46 In the arguments below we combine the previous cases depending on the possi

ble 3rd and 4th rows. For example, Case 1.1.1-1.2.1 means the 3rd row follows Case

1.1.1 and the 4th row follows Case 1.2.1. When x ^ 0, we study the possibility of a combination with different 3rd and 4th row vectors with the combination parameters

{u\,Vi), (v.2, V2), respectively. From the discussion above, we have the following pos sibilities.

Case 1.1.1-1.2.1: f x 1 a a

1 X a 1

a a X * (tti, Vi)

a 1 • X {U2, V2)

From Case 1.1.1 and 1.2.1, we have a ^ 0, x2 + x — 2a2 = 0; and a —1,

(a: — a)(x2 + ax — 2) = 0.

So x = —2(1 + 0), a = , then s/h(n — s) = 2n, no such real numbers n and s exist. So this combination is impossible. By symmetry, the combination of Case

1.1.1-1.3.1 is also impossible.

47 Case 1.1.1-1.2.2:

X 1 a a

1 X a 1

a a X • («1, Vi)

a 1 •k y («2, V ) / 2

By the results in Case 1.1.1 and 1.2.2, we have a ^ 0, x2 + x — 2a2 = 0; and a ^ —1, x3 + (l — a)x2 — (a2 + 2)x + 3a — 1 = 0, which follows (a + 2)x = 2a2+ 2a — 1.

Since Ux(x + 1) = a, then Ui(a 4-1) = = a or 1, which implies x — a < 0 or x = a2 + a — 1, thus a — a2 = 0 or (a + 2)(a2 + a — 1) = 2a2 + 2a — 1, it follows a = 0, a = l,ora = —1 which is a contradiction. So this combination is impossible.

By symmetry, Case 1.1.1-1.3.2 is also impossible.

Case 1.1.1-1.4.2:

X 1 a 1

1 X a 1

a a X * (Ui, vx)

• (U2, V ) 1 1 V 2

By the results in Case 1.1.1 and 1.4.2, we have a ^ 0, x2 4- x — 2a2 = 0, and x2 + (2 — a)x — (a + 1) =0, it implies that x = 2a + 1 and a = :y^r5. Since si == S4 = 1, x — 9^, y = |(1 — a), we have a = — a contradiction. So this combination is impossible. 48 Case 1.1.2-1.2.1:

X 1 a a

1 X a 1

a a y • (ui, Vi)

a 1 • X / ('u2, v2)

By the results in Case 1.1.2 and 1.2.1, we have x = 2a — 1, y = a, a € (0, |) or x = — 1 — a < 0, y = -2a > 0; a ^ —1, (x-a) (a:2-fax — 2) = 0. When x = a, we have a = 2a — 1, a = 1, a contradiction or a = —a — 1, then x = a = — |, y = 1, s = 3p; when a:2 + ax — 2 = 0, if x = 2a — 1, it follows a = 1 or a = — |, a contradiction; if x = —1 — a, it follows a = 1, a contradiction. We know the combination is possible for Case 1.1.2 and 1.2.1 when x — a = —|, y — 1, s — By symmetry, there is a combination of Case 1.1.2-1.2.1-1.3.1. By simultaneous row and column permuta tions, we get the Mirsky matrix of the form

/ \

'SI 2*/six(s2+l) "2^six(s3+l)

1 (3.32) " 2*^(»2+l)XSl 2"^»22>/s2+l + ^(*2+1)X(S3+1) 1 J ^(s3+l)>XSJ >^(S3+L)X(S2+L) 2 S34-1 where 0 < Sj < n — 3, i = 2 or 3, Si = ?.

49 Case 1.1.2-1.2.2:

X 1 a a

1 X a i

a a y • (til, Vi)

a 1 • y (M2. V ) / 2

By the results in Case 1.1.2 and 1.2.2, we have a ^ 0, x = 2a — 1, y = a or x = —1 — a, y = —2a, and a ^ —1, x3 + (1 — a)x2 — (a2 + 2)x + 3a — 1 = 0.

Since in the case 1.2.2, s2 < 1, it forces si < 1, then x = — 1 — a, y = —2a, so

(a — 1)(a 2 + 4a + 1) = 0 follows from x3 + (1 — a)x2 — (a2 -I- 2)x + 3a — 1 = 0, then a = —2 + \/3, x = 1 — \/3, and we have n = -\/3(n — s), but there are no such real numbers n and s, it is a contradiction. So this combination is impossible. By symmetry, neither is the combination of Case 1.1.2-1.3.2.

Case 1.1.2-1.4.2: T X 1 a 1

1 X a 1

a a y * (til, Vi)

1 1 • y / (u2, V2)

By the results in Case 1.1.2 and 1.4.2, we have x = 2a — 1, y = a, a € (0, |) or x — — 1 — a, y = —2a, a G (-1, 0) and then y(x +1) = 2, it follows a = 1 or a = —1, obviously this combination is impossible. 50 Case 1.2.1-1.3.2:

f X 1 a 1

1 X 1 a

a 1 X • (iti, t/i)

1 a • y / («a, v2)

By the result in Cases 1.2.1 and 1.3.2, we have a ^ — 1, (x — a)(x2 + ax — 2) = 0 and x3+(l—a)x2 — (a2+2)x+Za— 1 = 0. Ifx = a, then (a—l)2(a+l) = 0, a contradiction; if x2 = 2—ax, we have 0 = x(2—ax)+ (l—a)(2—ax) — (a2+2)x+3a— 1 = (a2—a)x—o+l, so x = \ then x2 + ax — 2 = 4J — 1=0, a contradiction. So this combination is impossible. By symmetry, the combination of Case 1.2.2-1.3.1 is also impossible.

Case 1.2.1-1.4.2: \ X 1 a 1

1 X 1 1

a 1 x * (tii, i/i)

1 1 * y, («2, v2)

By the results in Case 1.2.1 and 1.4.2, we have a ^ —1, (x — a)(x2 + ax — 2) = 0 and x2 + (2 — a)x — (a + 1) = 0. If x — a then a2 + (2 — a)a — (a 4-1) = 0, a contradiction; if x2 + ax - 2 = 0 then x = — a = — a contradiction. So this combination is impossible. By symmetry, the combination of Case 1.3.1-1.4.2 is also impossible. 51 By the discussion above Case 1.2.1-1.3.2 and Case 1.2.1-1.4.2 (which are impossi

ble) and x ^ 0, so there is one possibility for Case 1.1.2-1.2.1-1.3.1 which has been illustrated as the first matrix in Theorem 3.2.2.

Case 1.2.2-1.4.2: \ X 1 a 1

1 X 1 1

a 1 y * (itl, Vi)

1 1 * y } («2, Va)

By the results in Case 1.2.2 and 1.4.2, we have a ^ —1, x3 + (1 — a)x2 — (a2 + 2)x +

(3a — 1) = 0 and x2 + (2 — a)x — (a + 1) = 0. It follows (1 + a)x = 2, obviously a contradiction and this combination is impossible. By symmetry, the combination of

Case 1.3.2-1.4.2 is also impossible.

Case 1.2.2-1.3.2:

X 1 a

1 X 1 a

a 1 y • («1, Vi)

1 a • y («2, V2)

By the results in Case 1.2.2 and 1.3.2, we have S2 = S3 = 1, and there is no other row 52 allowed, this implies that n = 4 and x = a ¥" —then we have 5a2 — 12a —1 = 0 by x3 + (1 — a)x2 — (a2 + 2)x + 3a — 1 = 0 then a = 6~^. Now

(ui + ux)(l + x) = 1 + a,

(ui — vi)(l — x) = 1 — a.

It follows that £2(3,4) = Ui + av\ can neither be a nor 1, a contradiction. So this combination is impossible.

Thus we have discussed all the possible combinations for Case 1. The remainder of the proof above follows by applying a similar strategy for each of the 6 cases in

(3.30). •

Recall that two matrices A, B, with the same order, are said to be D-similarii there is a diagonal matrix D with diagonal entries equal to 1 or -1 such that DAD = B.

When x = a = — 1, y = 1, n = 2s, we use the notion of D-similarity and find the structure of a Mirsky matrix of the form

( - J8 \

^ J3 Js J

We have performed some numerical experiments to find a Mirsky's matrix (see the algorithm in Appendix), and it seems that extensive random searching cannot find much more than the theory posted here.

53 3.3 Spread of Some Real Symmetric Matrices

3.3.1 The Form of Real Symmetric Matrices of Rank 2, with

Entries a or 1

In this section, we study the structure of rank 2 symmetric matrices with entries a or 1, a € [—1, 1).

Lemma 3.3.1 Let A € Sn-{a, 1}, a € [—1, 1), with rank(A) = 2. Without loss of generality, let the first and second row vector a, (3 be linearly independent. Then any remaining row 7 must lie in one of the following cases:

1. if — 1 < a < 1, a 7^| — and a ^ 0, 7 = a or 7 = /?;

2. if a = —1, then 7 = a, 7 = /?, 7 = —a or 7 = — (3;

3. if a = then 7 = a, 7 = /? or 7 = —a — /3;

4. if a — 0, then 7 = a, 7 = /? or 7 = 0.

Proof. Since rank(A) = 2, the nonsingular upper left principal submatrix 5 of order

2 will be one of cases below:

\ ( \ / \ / \ a 1 a 1 1 a a a

a l) 1 aJ <1 1; 1° 1J Case 1 Case 2 Case 3 Case 4

a ^ —1 a 7^ —1 a^O

Let 7 be an arbitrary remaining row vector in A which we may assume is the 3rd row by simultaneous row and column permutations, if necessary. Then 7 = ua+v(3, u, v € R. 54 We show the details for Case 2 in (3.33), similar arguments can be applied for the remaining cases.

Case 2: \ a 1 ua + v u

1 1 u + v V

ua + v u + v u2a + 2uv + v2

The first principal submatrix of A, order 3, is shown above, and (u, v) is the pair of combination parameters for the 3rd row.

ua + v = 1 =r> (a — 1 )u = 0, u = 0, v = 1. u + v = 1

ua + v = a => (a — 1)M = 0, u — 0, v = a, thus >1(3,3) = a2, { u + v = a If a2 = 1, then a = —1, u = 0, v = —1; if a2 = a, then a — 0, u = 0, v = 0.

55 If

ua + v = a => (u - l)(a — 1) = 0, u = 1 and v = 0. u + v

ua + v = 1 => (u + l)(o — 1) = 0, u = — 1 and v = a 4-1, A(3,3) = u + av. { u + v — a If u + av = 1, a2 + a — 2 = 0, then a = 1 or -2, a contradiction; if u + av = a, then

a = — 1, u = —1, v = 0.

We study all the possibilities for 3rd row in A, and then the combinations of these

possibilities in case 2. Applying this argument for the remaining cases, we can draw

out the remaining conclusions in Lemma 3.3.1. •

Note: By Lemma 3.3.1, when — 1 < a < l,a ^ — |,a ^ 0, and rank(A) = 2,

then the remaining row 7 = a or /?, we let the numbers of rows equal to a and /? be

s and (n — s), respectively (1 < s < n — 1). Then the structure of A € Sn-{a, 1} is shown as below (up to simultaneous row and column permutation):

56 / \ / \

Type 1

(3.34) / \ / \ dJa (lJsx(n—s)

Type 3 Type 4

Here Type 1 arises from Case 1 in Lemma 3.3.1, Type 2 from Case 2, Type 3 from

Case 3 and Type 4 from Case 4.

When a = —^rank(A) = 2, we let the first and second row vector a, (3 be linearly independent. Thus, any row 7 is equal to either a ox (3 for Case 2 in Lemma 3.3.1, and then we can get Type 2 in (3.34), if the number of rows a and (3 is s,n — s respectively (1 < s < n — 1).

For Case 1, Case 3 and Case 4 in Lemma 3.3.1, since 7 = a,7 = /? or 7 = —a — (3. let the number of rows equal to a, f3 and —a — (3 be s, t and n — s — t, respectively

(1 < s < n — 1,1 < t < n — 1, s + t < n), we can verify the structure of A, by simultaneous row and column permutations, illustrated below:

57 1 JT -2 J3 'axt '5 «x(n(n—a—t) 1 T Hxa s—i) IT y 2 J(n-a-t)xa 2 J(n—a—t)xt J(n—a—t) y

Type 5

I R 1 T " 2 sxt 2 "sx(n-s-i)

— I J7 i j 2 txs Jt 2 "tx (n—s—t) (3.35) 1. T ^ 2 "(n—a-t)xs 2 J(n~a-t)xt J(n-s-t)

Type 6

\ _l / -- ax 2

Type 7

When a = —l,rank(A) = 2, without loss of generality, let the first and second row vectors a,0 be linearly independent. Then any remaining row 7 is equal to either a, 0, —a or —0.

For Case 2, 7 = a,0, —a or —0, let the number of rows equal to a,0, —a and —0 be s,t,si and £1, respectively (1 < s < n — 1,1

lustrated below:

( \ Ja JsX3\ Jaxt Jaxti

Jsixa Jsi Jsixt Jsixti

Jtxa Jtxai Jt —Jtxti

"JtlXS JtlXSl "JtlXt Jt\ which is D-similar to / \ -J,s+si J(s+si)x(t+t\)

(3.36) ^ <^(t+ii)x(s+si) Jt+ti / Type 8

For Case 4, we arrive at the same structure of A as Type 8, up to D-similarity and simultaneous row and column permutations.

When a = 0, rank(A) = 2, without loss of generality, let the first and second row vector a, 0 be linearly independent. Then any remaining row 7 is equal to either a, /? or 0. Take the number of rows equal to a, /3 and 0 be s, t and n — s — t, respectively

(1 < s < n — 1,1 < t < n — 1,3 + t < n). The structure of A then follows, by simultaneous row and column permutations, and is illustrated below:

59 f \ 0«x(n—s-t) «/»xt

0(n—s—t)xs On—g—t 0(n-a—t)xt

^ Jtxs 0tx(n—s-t) 0, y

Type 9

/ \ 0s Osx(n—s—t) •/sxt

0(n—s—t)xs On—s—t 0(n—s—t)xt

^ ^txs Otx(n—s-t) Jt y

Type 10

On—a—t 0(n_s_t)xs 0(n-s—t)xt

Osx(n—s—t) Js 0sxt

^ Otx(n—s—t) 0*xs Jt

Type 11

60 3.3.2 The Spread of Real Symmetric Rank 2 Matrices with

Entries a or 1

In the previous section, we studied the structure of real symmetric rank 2 matri ces with entries a or 1, and now we focus on the maximum spread for these kinds of matrices.

Theorem 3.3.2 Let A be a real symmetric n x n matrix, rank 2, with entries a or

1, that satisfy —1 < a < 1, a ^ a ^ 0, n > 2. Then the following holds.

1. If the structure of A is Type 1 in (3.34), then

n if n is even, SP(A) < " \/n2 + a2 — 1 if n is odd.

If n is even, equality holds if and only if s = If n is odd, equality holds if and only if s = or

2. If the structure of A is Type 2 in (3.34), SP(A) < When s=round(-^), the spread of A is maximum. (Here round(x) means an integer number closest to x.)

3. If the structure of A is Type 3 in (3.34), SP{A) < >An 2) +^° 1)+n^ equality holds if and only if s=l or s=n-l.

61 4- If the structure of A is Type 4 in (3.34),

2ny/a3/(3a + 1) if — 1 < a < —g,

y/n2 + 2(2a 2 — a — l)n — (3a2 — 2a — 1) if - \

2 SP(A) < i yfn — 8n/9 */ a=~3>

y'n2 + 2(2a2 — a — l)n — (3a2 - 2a — 1) z/ - g < a < 0,

(a + n- l + -^n2 + 2(2a2 - a — l)n - (3a2 — 2a- l))/2 i/ 0 < a < 1.

Proof. For Type 1, we have / A

and if we let /(A) = det(XI — A) = 0 be the characteristic polynomial of A. Then

/(A) = 0 can be rewritten as An_2[A2 — na\+ s(n — s)(a2 — 1)] = 0. Since s(n — s)(a2 —

1) < 0, we find

SP(A) — y/ (na)2 — 4s(n — s){a2 — 1) = l^a^s+^a2, and since 1 — a2 > 0, so SP(A) < n. Furthermore, if n is even, equality holds if and only if s = whereas if n is odd, SP(A) < Vn2 + a2 — 1, equality holds if and only if s = 2^ ors = In any event, we have SP(A) < n as a2 < 1.

For Type 2, we have ( \ Q,JS Jsx(n—s) A

62 Then /(A) = 0 can be rewritten as An 2[A2 — (as + n — s)A + s(n — s)(a — 1)] = 0.

Since s(n — s)(a — 1) < 0, we find

SP(A) = y/(as + n — s)2 + 4s(n — s)(l — a) = -y/(a + 3)(a — l)s2 + 2n(l — a)s + n2,

and (a+3)(a — 1) < 0, so SP(A) < yj^n. In this situation, we know h(a) = 3^ is a

monotone decreasing function. So 1 < y/h(a) < \/2. This implies that the maximum spread for matrices of Type 2 is bigger than the maximum spread for matrices of

Type 1, for the same value of a.

For Type 3, we have ( \ tJ (lj x[n—s) A = s s

&J(ri—s)xs Jn—i

Then /(A) =0 can be rewritten as An 2[A2 — nX + s(n — s)(l — a2)] = 0. By the conditions s(n — s)(l — a2) > 0 and n > 0, we find

n2 2 2 2 2 2 bF{A)ANT A\ =_ V ~ 4(n — s)s(l_ — a ) + n _ \/4(l — a )s — 4n(l_ — a )s + n + n

Obviously, SP(A) is maximum when s = 1 or s = n—1 and SP(A) < V^n ^+"

2 -2 + w 1 n 2 as g(a ) = V^" ) ^° ( ~ )+ jg a monotone increasing function and g(a ) < n. In this event, we have that the maximum spread for matrices of Type 2 is bigger than the maximum spread for matrices of Type 3, for the same value of a.

63 For Type 4, we have ( \ CtJ &Jsx(n—s) A = s \ &J(n—s)xs Jn—s

Then /(A) = 0 can be rewritten as A" 2[A2 — (sa + n — s)n\ + s(n — s)a(l - a)] = 0.

If a < 0, then s(n — s)a(l — a) < 0, and

SP(A) = y/(sa + n — s)2 — 4s(n — s)a(l - a).

If a > 0, then s(n — s)a(l — a) > 0 and sa + n — s > 0, then

onr,n _ -so + n-s + y/(sa + n- s)2 - 4 s(n - s)a(l - a) or {A) — - •

Furthermore, we find that:

When — 1 < a < — it implies that € (0, |). So SP(A) is maxi mum if and only if s = round(^^n) and SP(A) < 2nyjg^y. In this situa tion, we need to compare hi(a) = to h(a) = -1 < a < — since

4 1 3a h(a) — hi (a) = ^ (a^.3)(^.^ '* > 0, which implies that for the same a E (—1,—|), the maximum spread for matrices of Type 2 is bigger than the maximum spread for matrices of Type 4.

When — |

SP(A) < y/n2 + 2(2a2 - a - l)n - (3a2 - 2a - 1). 64 When a — —2n(l — a)(l + 2a) < 0. So SP(A) is maximum if and only if s = 1 and

SP(A) < yn2 - y < n.

When —| < a < 0 we have € (n, oo). So SP(A) is maximum if and only if s = 1 and

SP(A) < x/"2 + 2(2a2 - a - l)n - (3a2 - 2a - 1).

Define the function

2 2 2 h2(a) — \/n + 2(2a — a — l)n — (3a — 2a — 1)

= -*/(4n — 3)a2 — (2n — l)a + (n — l)2, —^ < a < 0.

2 Then h2(a) is a monotone decreasing function. Hence h2(a) < h2{—\) = yn —| < n. In this event, we have that for the same a € (—|,0), the maximum spread for matrices of Type 2 is bigger than the maximum spread for matrices of Type 4.

When 0 < a < 1, € (f n, n) and s(a — 1) < 0. So SP(A) is maximum if and only if s = 1 and

SP(A) < a + 71 ~ * + V^n2 + 2(2o2 - a- 1 )n - (3a2 - 2a - 1)

Define the function /13(a) = a+n~1+V'n +2(2a then /13(a) < n when 0 < a < 1, which implies that for the same a 6 (0,1), the maximum spread for matrices of Type 2 is bigger than the maximum spread for matrices of Type 4. •

65 When the entry a = — we have the following theorem on the maximum spread of the matrices with rank 2 in Sn-{a, 1}.

Theorem 3.3.3 Let A be a real symmetric nxn matrix of rank 2, with entries 1 or a = — Then the following holds:

1. If the structure of A is Type 5 in (3.35), SP(A) < n. Further, equality holds if and only if s = t.

2. If the structure of A is Type 2 in (3.34), then SP(A) < y^n. When s=round(^), the spread of A is maximum.

3. If the structure of A is Type 6 in (3.35), SP(A) < n+Vn Further, equality holds if and only if s = l,t = n — 1 or s = n — 1,< = 1.

4- If the structure of A is Type 7 in (3.35), SP(A) < n. Further, equality holds if and only if n — t = 2s.

Proof. For Type 5, we have (

-2 Js Jsxt -U2"sx(ri-3-t)

A = — I J ^ T Jtxs 2Jt '2"tx(n-3—t)

IT 1 j 2 "(n-s-()xs 2 "(.n-s—t)xt Jn-s—t 66 Then /(A) = 0 can be rewritten as An 2[A2 + 3a+32 2n A + f ((s +1)(s +1 — n) — st)] = 0.

Since (s + t)(s +1 — n) — st < 0, then

2 2 2 pr)/M I(3s + 3t — 2n) or/ , _ ; ~ /-3(s-i) +4n 5P(i4) = yi- 3[(s + t)(s + t — n) — st] = yj —* ^ .

SP(A) is maximum when s = £,l

For Type 2, by the related characteristic polynomial function, /(A) = 0, we have

SP(A) < yj\n. The maximum spread of A attains when s = round(^) and it is bigger than the maximum spread for matrices of Type 5.

For Type 6

— I T 1^ T Ja 2 axt ' 2 Jsx(n-s-t) _ 1 / 1 T 2"txa Jt 2 tx{n-a—t)

1. j 1 T 2 (n—s—t)xt Jn-a—t

Then /(A) = 0 can be rewritten as A" 2[A2 — nX + |((s + t)(n — s — t) + s£)] = 0.

Since (s + t)(n — s — t) + st > 0, then

2 2 SP(A) = ^(n +\Jn — 3n(s +1) + 3(s +t) - 3st). it

Define Fi(s, t) = n2 — 3n(s +1) + 3(s +1)2 — 3st be a function of s, t, which axe two integer numbers in [1, n — 1], s +1 < n. Then we have

^ = —3n + 6(s + t) — 3s = 0 (3.38) OF, at —3n + 6(s +1) — 3t = 0. 67 From (3.38) we have s = t = This is the only extreme minimum value of Fi(s, f), and the maximum value is attained on the boundary of a region formed by lines s > l,t > 1, s+ t < n. Since s, t are symmetric in Fx(s, t), it's convenient to assume s < t, and s + t = n on the boundary. This implies that if Fi(s, t) = n2 — 3st < n2 — 3(n — 1), then SP{A) < |(n + y/n2 — 3(n — 1)). Equality holds if and only \i s = \,t — n — 1 or s = n — l,t = 1. It is easy to see that \{n + y/n2 — 3(n — 1)) < n and it also shows that the maximum spread for matrices of Type 2 is bigger than the maximum spread for matrices of Type 6.

For Type 7

_I j 2 s -u2"sxt J,sx(n-a-t) A = 1 / '2"tus Jt 2hJ "tx(n-s-t)t

^ J(n-s-t)xs 2*^(n_a_f)xt _I2"n-s—t 7

Then /(A) = 0 can be rewritten as An 2[A2 — |(31 — n)A + |(s2 — (n — t)(s +1))] = 0.

Since s2 — (n — t)(s +1) < 0, then

2 2 SP(A) = n) - 3[s -(n-t)(s + *)].

2 2 Define F2(s,t) = |(3t — n) — 3[s — (n — t){s + t)] be a function of s, t, which are two integer numbers in [l,n — 1] with s + t < n. Then we have

^ = -6s + 3(n -t) = 0 (3.39)

dt = |(32 1 — n) + 3(n — s — 2t) = 0.

68 Prom (3.39) we have n — t = 2s. It is the only maximum value of f(s,t). So

SP(A) < n, and equality holds if and only if s — t. Then it shows that the maximum spread of matrices for Type 2 is bigger than the maximum spread of matrices for

Type 7. •

In the following theorem, it's convenient to denote s, t as s + si,t +1\.

Theorem 3.3.4 Let A be a real symmetric nxn matrix of rank 2, with entries 1 or a = — 1. If the structure of A is Type 8 in (3.36), then

y/2n if n is even, SP(A) < < y/2n2 — 1 if n is odd.

Equality holds if and only if s = t = when n is even; s = t = or s = 2#,t = when n is odd.

Proof. Since

"37 "SXt7 ^

'tX3 "t

Then /(A) = 0 can be rewritten as An_2[A2 4- (s — t)\ — 2st) = 0. Since st > 0, then

SP(A) = \/(s — t)2 + 8st = V—4s2 + 4ns + n2.

So SP(A) < y/2n. Further, when n is even, equality holds if and only if s = t = and when n is odd, equality holds if and only if s = 2=^, t = ^ or s = 2^-, t = and SP(A) = y/2n2 — 1. In fact, by Theorem 2.2.3, we know the conclusion holds since ||i4||ir = n in this situation. • 69 Theorem 3.3.5 Let A be a real symmetric nxn matrix of rank 2, with entries 1 or a = 0. Then the following holds.

1. If the structure of A is Type 9 in (3.37), then

n if n is even, SP(A) < < y/n2 — 1 if n is odd. n^l Equality holds if and only if s = t = when n is even; s — 9 i Lt or s = 2^, t = when n is odd.

2. If the structure of A is Type 10 in (3.37), then SP(A) <

3. If the structure of A is Type 11 in (3.37), then SP{A) < n — 1.

Proof. For Type 9, we have / \ 0a 0sx(n-s—t)

^ 0(n—a—t)xs On—s—t 0(n—s—t)xt

^ Jtxs 0tx(n-a—t) Of y

Then /(A) = 0 can be rewritten as An_2(A2 — st) = 0. Since — st < 0, then

SP(A) = 2Vst < (s + t)2 < n2.

Obviously when n is even, equality holds if and only if s = t = and when n is odd, SP(A) is maximum if and only if s = n^-,t = or s = = 2yi. The 70 maximum value is \fri2 — 1. It shows that the maximum spread for matrices of Type

2 is bigger than the maximum spread for matrices of Type 9.

For Type 10, we have /

A 0(n—a—t)xa On—s—t 0(n-s—t)xt

Jtxa 0tx(n—s—t)

Then /(A) = 0 can be rewritten as An_2(A2 — tX — ts) = 0. Since—st < 0, then

SP(A) = \/t2 + 4st. Let s+t = m < n, and define /2(s) = t2+4st = —3s2+2ms+m2,

2 2 where f2(s) is a function of s. Obviously, f(s) < |m < |n , equality holds if and only if m = n,s = So when s = round(^),t = n — s,SP(A) is maximum. This shows that the maximum spread for matrices of Type 2 is bigger than the maximum spread for matrices of Type 10.

For Type 11, suppose l<5

Then /(A) = 0 can be rewritten as An~2(A - t)(A — s) = 0. Since — st < 0, then

SP(A) = t < n — 1. Equality holds if and only if s = 1, t = n — 1. It shows that the maximum spread for matrices of Type 2 is bigger than the maximum spread for 71 matrices of Type 11.

Prom the theorems in this section, we know that the maximum spread is attained by some matrices of Type 2.

From some numerical experiments, we verify the above four theorems and find for any rank 2 matrix A € »S„-{a, 1}, a € [—1, 1), when its structure is same as that in

Type 2, then its spread is maximum. The followings are the examples for n = 4 and

Examples:

• n = 4 : / -0.9 -0.9 1.0 1.0 -0.5 -0.5 1.0 1.0

-0.9 -0.9 1.0 1.0 -0.5 -0.5 1.0 1.0

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

if a = -0.5, then s =

5.0000.

0.1 1.0 1.0 1.0 0.0 1.0 1.0 1.0

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 \ / \ if a — 0.1, then s = 1,5^(^4) = 4.5177; if a = 0.0, then s = 1,SP(A)

4.5826. 72 • n = 5 : ( \ 0.0 0.0 1.0 1.0 1.0

0.0 0.0 1.0 1.0 1.0

1.0 1.0 1.0 1.0 1.0 .

1.0 1.0 1.0 1.0 1.0

^ 1.0 1.0 1.0 1.0 1.0 ;

if a = 0.0, then 5 = 2, P(A) = 5.7446.

\ -1.0 -1.0 1.0 1.0 1.0

-1.0 -1.0 1.0 1.0 1.0

1.0 1.0 1.0 1.0 1.0 >

1.0 1.0 1.0 1.0 1.0

if a = —1.0, then s = , SP(A) = 7.0000.

< \ -0.5 -0.5 1.0 1.0 1.0

-0.5 -0.5 1.0 1.0 1.0

1.0 1.0 1.0 1.0 1.0 >

1.0 1.0 1.0 1.0 1.0

1.0 1.0 1.0 1.0 1.0 /,

if a = —0.5, then s = , SP{A) = 6.3246.

73 3.4 Conclusion about Real Symmetric Matrices

3.4.1 Maximum Spread of Real Symmetric Rank 2 Matrices

with Entries in the Interval [a, 1] ( — 1 < a < 1 )

From the work in Section 3.3, we arrive at our main contribution involving the maximum spread for rank 2 matrices in 5„[a, 1].

Theorem 3.4.1 Let A € 5n[a, l],a G [—1,1) and rank(A) = 2, if SP(A) is maxi mum, then A must be in the following form (up to permutation similarity):

(ITa J so "sox(n—so) \ A = ^ J(n-so)xso Jn-so y and

s0 = round(^)

{SP(A) = y/(a2 + 2a — 3)SQ + 2n(l — a)so + n2 .

Theorem 3.4.2 If the symmetric real matrix is ( \ aJason Jsox(n-so)J, A = (3.40) J(n-so)xso Jn-so

T T with unit vectors X = (xi,..., xn) ,Y = (yi,..., yn) being the eigenvectors corre sponding to the eigenvalues Ai,A„ of A, respectively, then

XiXj - ytyj ^ 0,i,j € {1,2... ,n}. 74 Proof. Since det(XI — A) = 0, we know Ai, An must satisfy

2 a _ n A - (n + (a — l)s0)A + ( l)( ~ -So)so = 0, so

= Al An fi (a — 1)^0

AxAn = (a - l)(n - s0)s0, and AX = X\X, AY = ANY. We can rewrite these equations as AZ = XIZ, Z = X,

T if i = 1; Z = y, if i = n, where Z = (zx,..., zSo, zSo+1,..., zn) , f < s0 < §, n > 2.

Thus 50 aYlzt+Ylzt== Ai2m'm = -1, • • •' s° 1 «0+l and so n ^2Zt + Zt = ^iZr'r ~ s°+••• > "• 1 ao+l

Since Ai > 0 > An it follows that

= Zi • • • — ZSq, ZSQ+i = • • • — zn,

AiZn = (Ai + (l-o)a0)«i,

Aj + (1 — a)so > 0.

If A„ + (1 - a)s0 = 0, then yn = 0, which implies yx = 0 by s0j/i + (n - s0)y„ = A„y„.

Thus Y = 0, a contradiction. Since ||Z||2 = 1, we have

2 A? A, + (1 — a)s0 1 2 11 s0A? + (n - s0)(Aj + (1 - a)s0) ' " A<

75 and

VT1 V , i vAi + (l — A)«o An + (1 —a)so /Q;in X Y = SoXiyi + (n - s0) Xj yx (3.41) Al An

(Al +(1 a)s K +(1 n)So) - xlVl(<>0 + ~ °>[ " (n - ,„)) = 0. (3.42) AiA„

Without loss of generality, we suppose Xi, y\ > 0, then we prove xtxj — ytyj ^ 0 in different cases:

(1) i, j E {l,...,s0};

(2) i, j € {s0 +

(3) i e {1,..., s0}, j G {so + 1,..., n} or j 6 {1,..., s0}, i € {s0 + 1,..., n}.

In case (1):

x&j - yiyj =x\-y\

= M ^ s 2 2 2 oA + (n — So)(Ai + (1 — a)s0) + (n — so)(An + (1 — a)so) ' because

2 2 2 2 Ai(^oA^ +(n — So)(An + (1 — a)so) ) — A (soA + (n — so)(Ai + (1 — a)so) )

2 2 = (n — So)[(AjAn + Ai(l — a)so) — (AiAn + Ai(l — a)so) ]

= (n — so)(l — a)so(Ai — An)(2AiAn + (1 — a)so(Ai + An))

2 2 = -(n - s0) (l - a) So(Ai - An) < 0,

we have XiXj — ytyj 0.

76 In case (2):

2 ~ ViVj = x n~yl

2 2 (Ai + (1 - a)s0) _2 (An + (1 - a)s0) 2 A2 Xl \l Vl 2 2 (Ai + (1 — a)sp) (An + (1 — Q)SQ) 2 2 2 s0Af + (n - s0)(Ai + (1 - a)s0) s0A + (n - s0)(An + (1 - o)s0) because

2 2 2 (Ai + (1 - a)s0) (soA + (n - s0)(A„ + (1 - a)s0) )

2 2 2 ~ (An + (1 — a)so) (5oA + (n — So)(Ai + (1 — a)so) )

= 2 — 2 So[(AiAn + A„(l — a)so) (AiAn + Aj(l — a)so) ]

= s0(l — a)so(An — Ai)(2AiAn + (1 — a)so(Ai + An))

2 = SQ(! - a) (An - Ai)(so(l + a) - n),

since ^ < |, 1 + a < 2, then s0(l + a) < n and XiXj — yii/j ^ 0.

In case (3): by symmetry, without loss of generality, suppose i € {1,...,so}, j €

{s0 + 1,... ,n}, then

XiXj ViVj — %n VlVn

_ Ai + (1 — a)s0 _2 A„ + (1 — a)s0 2 — \ xi \ Vi> Ai An

77 because

2 (Aj + (1 — a)soAi)(soAjl + (n — so)(An + (1 — a)so) )

2 2 " (A + (1 - a)s0An)(s0A? + (n - s0)(X1 + (1 - a)s0) )

= (1 — a)so(An — Ai)[soAiAn — (n — so)(2AjAn + (1 — a)so(Ai + An))

2 + (n - 50)(AiAn - (1 - a) So)]

2 A = 2(n - s0)(l ~ a) «o( i ~ An), we have XiXj — yiyj ^ 0. •

78 3.4.2 Maximum Spread of Real Symmetric Matrices with

Entries in the Interval [a, b]

In the previous section, we studied the case, a real symmetric matrix A with entries in an interval and rank(A) = 2. We know that if rank(A) = 1, then the related spread can not be maximum. We have performed many numerical experiments (see the Appendix) to find the maximum spread for a real symmetric matrix, and the results of these experiments suggest the following conjecture.

Conjecture 3.4.3 For the class of real symmetric matrices A with entries in the interval [a, 6], the maximum spread in the class must be attained by a real symmetric matrix Sn-{a, b} ofrank(A) = 2.

Case 1: If a2 < b2, let u — ab'1. Then we have that the maximum value SP(A) of spread

b\/ (u2 + 2u — 3)s2 + 2n(l — u)s + n2, where s = round{^3), and by row and column permutation the structure of A is given by ( \ aJs bJSx(n—s)

6J(n-s)xs bJn-g J

Case 2: If a2 > b2, let u — ba-1, then we have that the maximum value SP(A) of

79 spread

—ay/ (u2 + 2u — 3)s2 + 2n(l — u)s + n2, where s = round(^5), and by row and column permutation the structure of A is given by ( bJs Q"Jsy.{n—s)

Q-J(n—s)xs O'Jn-s \ We have tried to compare the spreads between the real symmetric matrices with entries in an interval of rank more than or equal to 2. And even for the rank 3 case, we can not determine sufficient information about the distribution of the eigenvalues to derive any definitive conclusions. For example, let / \ ( \ a 1 1 CL € 1 1

1 1 1 and its perturbation A{e) = 1 1 - e 1 1 1 1 \ 1 1 1 - e where a € [—1,1) and e > 0. So now rank(A(e)) = 3, but it is complicated to compare its spread with SP(A) since the characteristic polynomial of A(e) is a cubic function of e and it is difficult to describe the associated spread of A(e). We know that SP(A(e) < SP(A) + 2e by Weyl's Perturbation Theorem [30], but we can not prove that SP(A(t)) < SP(A).

80 3.5 Real Skew Symmetric Matrices

In this section, we let .SSnf—a, a] be the set of real skew symmetric matrices with entries in the interval [—a, a]. It is well known that for a real skew symmetric matrix, all of the nonzero eigenvalues are purely imaginary, and thus are of the form iAi, — iAi, i\2, — i\2, • • •) where each A* is real.

Theorem 3.5.1 Suppose A € 55n[—a, a] and rank(A) = 2. If spread of A, SP(A), is maximum, then each off-diagonal entry of A is either a or -a.

Proof. If rank(A) = 2, and suppose there are two conjugate complex eigenvalues with corresponding eigenvectors q\,q2- Let

/ 1 1 ? £

72 72 c X 1 U = Q i — i 72 V2 e

\ 0(n--2)x2 I(n-2)x(n-2) / Then Uj — -j%q\ + ^?2> u2 — ~ and / i\i 0 02x( n-2) U'AU 0 —i\i

y 0(n-2)x2 0(n-2)x(n-2) Then

T SP(A) = 2|Ai| = \u\Aui - u2Au2\ = |qfAq2 - <£Aq\\ = \e [A o faqj - q2qf)]e|.

Note that B = qiq% — q2qj is a skew symmetric matrix. Prom this we define a new rank 2 normal matrix A = (a^) as = a if quQ2j — quq\j > 0, otherwise ay = —a if quq2j ~ Q2iqij < 0. It follows that SP(A) < SP(A). • 81 By considering a XA instead of A, we need only study the maximum spread in

SSn[-1,1].

Notes:

1. Let F G {R, C}, and X G F be a convex and compact set. Define

Jifn(X) = {a = [ai-?] G M„(F) | a* = a & ai:?- G X,Vi, j},

yjifn(X) = {b= [Ai] G M„(F) |b* = -b & fa G X,Vt, j}.

Define the exterior, extX, as the set of all exterior points of X, then we have

the following proposition.

Proposition 3.5.2 J4fn(X) & S?Ji?n(X) are both compact and convex sets.

Moreover,

a G extJifn{X) a^ G extX,Vi,j

Scb G ext^Jffn(X) Pij G extX,Vi,j.

The proof is straightforward. •

2. For a G Jifn(X), let A*(a) = n-tuple of eigenvalues of a, in descending order. So,

A1 (a) G RN.

Recall Weyl's Perturbation Theorem: VHermitian a, b G Mn(F),

max |Aj(a) — Aj(6)| < ||a — 6||, where || • 11 is an operator norm.

Hence, the map Aj : Jf^(X) —> M is continuous, VI < j < n. In particular, the

spread SP = A| — A* = Xmax — Xmin is continuous. 82 As, J%(X) is convex, it is path connected, and thus,

{5P(a) | a G J4?n(X)} is a closed interval in R.

Because yjt?n(X) = iJ4?n(—iX), the same ideas apply, but we do need to take

into account that the diagonal entries are all equal to zero. That is,

Proposition 3.5.3 {SP(a) |a e J%(X)} and {SP(b)| b € ^J(fn(X)} are

closed intervals.

3. A function / : C —> R, where C is a convex set in a vector space, is

(i) convex, if f{tx + (1 -t)y) < tf(x) + (1 -t)f(y),Vx,y e C,t e [0,1]

(ii) concave, if — / is convex.

Note that Xmax : Jt^(X) —• R is convex, and Amin : J4fn(X) —> R is concave.

Hence, the spread SP : J%(X) —* R is a convex function. By Bauer's Maxi

mum principle, the continuous convex function SP attains its maximum at an

extreme point of J^(X). Hence, if a € J%(X) satisfies SP(a) > SP(a'),Va' €

Jif^(X) then € extX,Vi,j. The above discussion offers another viewpoint

and proof of Lemma 3.2.1 and Theorem 3.5.1, and the above analysis was the

result of some communication from Dr. Douglas Farenick.

Now we focus on rank 2 matrices A 6 SS^-l, 1]. As before there must exist an invertible principal submatrix of order 2 in any real skew symmetric matrix of rank 2.

Assume that this submatrix lies in the upper left block by simultaneous permutation

83 of rows and columns of A. Therefore any other row must be a linear combination of the first two rows. As a result we can determine the structure of these matrices.

Examples for such matrices with a pair of conjugate complex eigenvalues are : / \ 0 0-1-1 ( \ 0 -1 -1 0 -1 0 0-1-1 An = 1 0 -1 A4 = 1 °/ 11 0-1 1 1 0 1110

84 3.5.1 Structure of Real Skew Symmetric Matrices Whose

Spreads Attain the Bound in the Mirsky's Theorem

It is evident that the structure of an invertible principal submatrix of a real skew symmetric matrix A = [a^] with off diagonal entries -1, 1 or 0 is the matrix Ai above.

Furthermore the spread of this matrix attains the bound in Theorem 2.2.3 when its rank is 2.

Now, if necessary, assume A2 to the upper left corner of A by row and column permutations and therefore the other row vectors of A must be linear combinations of the first and second rows of A. Let r\, r

Claim 3.5.4 If 0 < 7\, Tj < 1, then if and only if ^ = rj = 1 and |Qi — 0j| =

/ 2 kir ± fc G Z then |ay| = 1 and \\A\\ F attains the maximum value.

Claim 3.5.5 If 0 < rj < 1 < rj < s/2, then if and only if = 1, rj = \/2 and

18i — 9j| = k7t ± k € Z then |ajj| = 1 and ||i4||^. attains the maximum value.

2 2 2 2 2 Claim 3.5.6 If 1 < n < rj < \/2, \an\ + |ai2| + |flji| 4- |aj2| + |aij| <4 if and only if r{ = Tj = V2 and &i,9j are in collinear, equality holds. 85 Case 1 Case 2

Figure 3.1: Two cases of the combination parameters in real skew symmetric matrices

Figure 3.2: Case for both arguments in quadrant I

Proof. Without loss of generality, assume ^ in quadrant I, if dj is in III or IV, then

\dij\ keeps the same value as argument Qj is in I or II. There are then just two cases to consider.

Case 1: (at, /3i),(atj, (3j) are both in quadrant I. Then by the Figure 3.2 above, we

have

\a{j| = riTj sin(6>j - 0i) < 1 - y/(r? -l)(r] -1) < 1.

86 Subcase 1 Subcase 2

Figure 3.3: Subcases 1 and 2

Thus

I Oil I2 + |a»21 2 + |ajl|2 + |Oj2|2 + hjl

= r? + r* + 1 + (r? - l)(r| - 1) - 2^(r? - l)(rj - 1)

2 < 2 + r?rj - 2(rt - 1)

= 4 + rf (rj - 2)

<4,

and equality holds if and only if r* = rj = %/2.

Case 2: (at,pi) is in quadrant I, and (aj,(3j) is in II, and there are four subcases

total to consider.

Subcase 1: See Figure 3.3 above. If 0* is located in L\ and 6j in L3, then

hi I = rir: sin(dj -0i) = 1 + \J(rf ~ 1 )(*$ ~ !) > 1.

which contradicts the constraints that the absolute value of entries are at

most 1. 87 Subcase 2: See Figure 3.3 above. If di is located in L\ and dj in L4, then

K"l = yj(ri - !) + ]/(rj - !) ^

which implies that rj < Then

|Oil|2 + |«i2|2 + \ajl\2 + Iaj2|2 + Ia»j |2

= r? + rj + {yjrf — 1 + yjr? - l)2 5 <2 x- + 1 4 < 4.

Note: By subcase 1 and subcase 2, we have (dj - 0j) £ [|, 7r), and |

decreases as (0j — 0*) increases. When dj is located between L3 and L4, di

2 2 2 in Li. If dij = 1, then this forces r*i < rj < so \aix\ + \aa\ + |aji| +

2 2 |aj2| + |ay | = r? + rj + 1 < 4.

Subcase 3: If di is located in L2 and dj in L3, then

\an\ = \J(ri - 1) + yVj - !) ^ !.

which implies that r_, < as in subcase 2.

Subcase 4: If di is located in L2 and 9j in L4, then

kjl = 1 + \/(r? - l)(r] - 1) > 1,

which is a contradiction.

Note: By subcase 3 and subcase 4, we have (dj - di) € (0,tt), and |1

increases to 1, then is larger than 1, and then decreases back to 1, as (dj—di) 88 increases. When 9j is located between L3 and L4, 0* in L2, if a{j = 1 then

this forces that rj < so |aii|2+|ai2|2+|aji|2+|aj2|2 + |a»j|2 = r2+r2+1 <

Equality holds if and only if r< = rj = \/2 and 9i,Qj are collinear. In this

2 2 2 2 2 case we have |aa| + |ai2| + \o>ji\ + |aj2| + |a»j| = 4 and = 0, without

loss of generality we let = 6j = f. •

Theorem 3.5.7 If A = [a^] € 55„[—1,1], and rank(A) = 2, with |a^-| < 1, 1 < i,j < n, then SP(A) is maximum if and only if A is permutationally similar or

D-similar to ( \ Orxr Jrxs Jrxt

Jsxr Osxa Jsxt

\ Jtxr Jtxs Otxt where 1 < r, s < n and r + s + t = n, r, s, i 6 N. Moreover, in this case

•^(2n), ifn = 0(mod 3), max(SP(A)) = < &(2y/n2 — 1), i/n = l(mocf 3), or n = 2(mod 3).

To prove the above theorem, we need the following lemmas.

Lemma 3.5.8 Lei = [a

raaa:(jjj4|l>) = < 6k2 + 4k ifn = 3k + l,

6k2 + 8k + 2 if n = Zk + 2.

89 Proof. There is a principal submatrix £)-similar or permutationally similar to A2 when rank(A) = 2 set in the upper left corner of A. If ||y4||| attains the maximum value, and the remaining row vectors of A are linear combinations of the first two row vectors of A, by the claims 3.5.4, 3.5.5, and 3.5.6, the combination parameters are

(1,0), (0,1) or (1,1), and the related numbers of these parameters are si,s2, and s3, respectively, then A is D-similar or permutationally similar to / \ Olxai Jlxsi "AXS3

(3.43)

where s^ s2, s3 € [0, n - 2], sx + s2 + S3 = n - 2, n > 3.

So now

II-^IIF — 2[1 + SI + s2 4- 2S3 + SI x S2 + SI x S3 + s2 x S3]

— 2[—(si + S2)2 + (n — 3)(si S2) + (2n — 3) + si x S2]

3 ^ —2 + S2)2 + 2(n ~ + + (4n ~ 6)

< 4rc — 6 + ^(n — 3)2. o

90 It follows that

6k2 =| n2 if n = 3fc,

and Si = s2 = k — 1, s3 = fc,

6A;2 + 4& = |(n2 — 1) if n = 3A: + 1,

and Si = s3 = k, s2 = k — 1;

or Si = k — 1, s2 = S3 = k; max(\\A\\p) =

or Si = S2 = k — 1, s3 = k +1,

6A;2 + 8k + 2 = |(n2 — 1) if n = 3A; 4- 2,

and Si = k,s2 = k — l,s3 = k + 1]

or Si = k — 1, s2 = k, S3 = A; + 1;

or Si = s2 = S3 = k.

By necessary D-similarities or permutation similarities, from (3.43) A is of the form / \ Orxr Jrxs Jrxt A = Jsxr Osxs Jsxt

Jtxr Jtxs Otxt V where 1 < r, s < n,r + s +1= n.

Claim 3.5.9 If |a| < 1, \/3\ < l,\a - (3\ < l,a,(3 € R, then a2 + (32 + (a - (3)2 < 2.

By the claims above equality holds if and only if the combination parameters (a,(5) =

(1,0) or (0,1) or (1,1). 91 Lemma 3.5.10 Let An = [ajj] € S5n[—1,1], rank(An) = 2. Then SP(A) is max imum if and only if there is a 2x2 invertible principal submatrix of A which is

D-similar to A2. After necessary row and column permutations, assume Ai is in the upper left corner of A. Then any other row vector of A is a linear combination of the first two row vectors vrith combination parameters (ai,0i),i = 3, By the claims above, these parameters are of the form (1,0), (0,1), (1,1) and let the number of these parameters are Sj, S2, S3, respectively. Then

Sl = s2 = k -l,s 3 = k, if n = 3k]

si = S3 = k, S2 = k — 1, or

Si = k — 1, s2 = s3 = k, or

Si = S2 = A: — 1, S3 = k + 1, if n — 3k + 1;

Si = k,S2 = k — 1, s3 = k + 1, or

Si = k — 1, S2 = k, S3 = k + 1, or

Si = s2 = s3 = k, if n = 3k + 2.

Proof. We use induction on n to verify Lemma 3.5.8. By the claims above, when n — 2, A is D-similar to A2 which has the maximum spread; and when n — 3, A is D-similar or permutationally similar to A3 which has the maximum spread; and when n = 4, A is D-similar or permutationally similar to A4 which has the maximum spread.

Now if A3k has the structure of matrix (3.43), when n increases to 3&+1, we assume the combination parameter for the last row vector is {a3k+i,03k+i) and also that

92 s1 = s2 = k — 1, S3 = k, k > 1, and si,s2,s3 € [0,3/: — 2], sa + s2 + S3 = 3k — 2, n > 3.

Then it follows that A$k+i is of the form \ 0 -1 OlXSi —J1XS3 —/?3fc+l

1 0 ^lx«i 0lx»a ^lxa3 <>3*:+1

OsiXl — J'sjXl X«1 Jsi XS2 "^llX»3 —03k+lJsiXl

Js^X 1 04jxl 0«2XJ2 CX3k+\Js3xl

•^S3Xl ~Js3x 1 •JaiXSl ~

— @3k+1 -<*3fc+i 03k+lJlxai Q3fc+l«^lx«a (/?3Jfe+l - a3fc+l)^lxa3 0 /

Now

= s s 2 II^W-IIIF ll^3fc|lF + 2[Pl k+i{ \ + 1) + a|fc+l( 2 + 1) + (a3fc+i — /?3fc+l) S3]

2 — 2 = 6fc + 2A:[a3fc+1 + /3ffc+1 + (c*3fc+i /?3fc+i) ]

< Qk2 + 4k.

This implies that when n = 3k+ 1,

s'j = k, s2 = k — 1, S3 = k, or

s[ = k — 1, s2 = k,s'3 — k, or

s'l = k — 1, s'2 = k — 1, S3 = k + 1,

where s[,s 2, S3, are numbers of combination parameters of (1,0), (0,1) and (1,1) in

Azk+l-

If has the structure of matrix (3.43), when n increases to 3& + 2, we assume the combination parameter for the last row vector is (a3k+2,03k+2)-

93 In this case if Si = k, s2 = k — 1, S3 = k, k > 1, then we have

= II^AM-IIIF + 2[/?3fc+2(Sl + 1) + a3fc+2(52 + 1) + (t*3fc+2 ~ fok+i)2S3]

2 a 2 = 6A: + 4fc + 2{ka\k+2 + (k + 1 )0ik+2 + fc( 3fc+2 ~ Pak+2) ]

2 = (6k + 4 k) + 2k[alk+2 + P>lk+2 + (a3fc+2 — Pzk+i)2) + Wtk+2

5; 6A:2 -4- 8k + 2.

Equality holds if and only if a3fc+2 = (33k+2 = 1 or a3k+2 = 0, /?3fc+2 = 1. This implies that when n = 3k + 2, s[ = k,s' 2 = k — 1, s'3 = k + 1, or S'J = s'2 = s3 = k, SP(A3k+2) attains the maximum possible value.

If si = k — 1, s2 = k, s3 = k, k > 1, we have

2 ||^3fc+2||F = ||-^3fc+l ||F + 2[0LK+2(S\ + 1) + Q!3FC+2(S2 + 1) + (a3fc+2 ~ PZK+2) S3]

2 2 2 = 6k + 4k + 2[(k + l)a:3fc+2 + kp k+2 + k(a3k+2 ~ /?3fc+2) ]

^ Qk2 + 8k -f- 2.

Equality holds if and only if a3k+2 = (33k+2 = 1 or a3fc+2 = 1,03k+2 — 0. This implies

that when n = 3k + 2, s[ = k - I, s'2 = k, s3 = k + 1, or s'j =s 2 =s' 3 = k, SP(A3k+2) attains the maximum possible value.

94 If Sj = s2 — k — 1, s3 = k 4- 1, k > 1, we have

S Q |M3*:+2|IF = |M3fc+l|lF + 2[/?ffc+2(Sl + 1) + <*3fc+2( 2 + 1) + ( 3fc+2 ~ Pzk+i)*$3]

2 2 = 6k + 4k + 2[ka\k+2 + Ar/?ffc+2 4- (A; 4- l)(a3jt+2 — /?3fc+2) ]

^ 6k2 4- 8k 4- 2.

Equality holds if and only if a3k+2 = 1, P-sk+2 = 0 or a3fc+2 = 0, /?3fc+2 = 1. This implies that when n = 3k + 2, s[ = k — 1, s'2 = k,s'3 = k + 1, or s[ = s'2 = s'3 = k, SP(A3k+2) attains the maximum possible value.

If A3fc+2 has the structure of matrix (3.43), when n increases to 3k + 3, we assume the combination parameters for the last row vector is (a3fc+3,03m)-

In this case if Si = k, s2 = k — 1, s3 = k 4- 1, A; > 1, then we have

S 2 ||^3fc+3||F ~ H^3fc+2||f" + 2[/?3fc+3(si + 1) + C*3fc+3( 2 + 1) + (<*3fc+3 ~ 03k+3) S3]

2 2 = 6fc 4- 8k 4- 2 + 2{ka\k+3 4- (A: 4- l)/?ffc+3 4- (k + l)(e*3fc+3 — /?3fc+3) ]

< (6k2 4- 8A: 4- 2) 4- 4(A; 4-1).

Equality holds if and only if a3k+3 = 0, fck+3 = 1. This implies that when n =

3k 4- 3, s'j = s'2 = kt s'3 — k + I, SP(A3k+s) attains the maximum possible value.

95 If Si = S2 = S3 = k, k > 1, we have

||^3fc+3|lF = + 2[/?3fc+3(sl + 1) + Q3Jfc+3(s2 + 1) + («3fc+3 ~ (hk+3)2S3]

= (6k2 + 8k 4- 2) + 2[(k + 1)0:3^+3 + (k+ l)Plk+3 + Ma3A:+3 ~ /?3fc+3)2]

< (6A;2 -h 8fc + 2) -h 4(fc H-1).

Equality holds if and only if ocsk+3 = @3k+3 = 1- This implies that when n =

3k + 3, s[ = s'2 = k, s'3 = k + 1, SP(Ask+3) attains the maximum possible value.

By the induction on the order of A, if SP(A) attains the maximum spread, then

A has the same structure as the matrix (3.43). It is evident that A is D-similar or permutationally similar to / Orxr Jrxs Jrxt

Jaxr Oaxs Jsxt

y Jtxr Jtxs Ofxt where 1

Qk2 =| n2 if n= 3k,

2 \\A\\ f 6k2 + 4k =| (n 2 — 1) if n = 3k + 1,

6k2 + 8k + 2 = |(n2 — 1) if n = 3k + 2.

So SP(A) <

96 3.6 Case of Complex Normal Matrices

In this section, we focus on two classes of complex normal matrices with entries of modulus at most 1. We use to denote the class ofnxn Hermitian matrices and yjtfn denote the class ofnxn skew Hermitian matrices.

3.6.1 Structure of Complex Hermitian Matrices Whose Spreads

Attain the Bound in Mirsky's Theorem

Let A = [ay] € J%(C), ||a,j|| < 1, if SP(A) attains the upper bound in Theorem

2.2.3 and n is even, rank(A) — 2 and tr(A) — 0, then there is a 2 x 2 invertible principal submatrix after necessary row and column permutations, / \ 1 a J \ 5 _1 / in the upper left corner of A. Thus A may be produced by the strategy that the remaining row vectors are combinations of the first 2 row vectors in A. Furthermore,

1, -1 must occur along the main diagonal of A in pairs. We can then derive the structure of A by studying its principal submatrix. Suppose the 4x4 principal submatrix A[{\,2,2m + 1,2m + 2};{1,2,2m + 1, 2m + 2}], 1 < m < ^ — 1, has the form / \ 1 a b d

a —1 c e

b c 1 / 1 t—'

^ d e *-*>1 97 Let the 1st, 2nd, (2m+l)th,(2m+2) th row vectors be a,f3, 7, 6 and 7,5 are combination of a, (5 with combination parameters (s, t), (p,q), respectively. It follows that

s + ta — b

sa — t = c (3.44) sb + tc = 1

sd + te = f.

By (3.44), we have t(cb — a) = 0, from which it follows that t = 0 or cb = a. If t = 0, then s = b, c = ab, f = db\ if cb = a, then c = ab,s = b,t = 0, / = db] and

p + qa = d

pa-q = e (3.45) pd + qe = — 1

pb + qc = f.

By (3.45), similarly we have p = 0, q = —e,d = —ae,f = db = —abe. This implies that -A[{1, 2, 2m + 1,2m + 2}; {1,2,2m + 1,2m 4- 2}] is of the following form, / \ 1 a b —ae

a -1 ab e

b ab 1 —abe

—ae e —abe —1 \ /

1 In other words, if a2m+i,i = b, then the (2m + l)" row vector of A is a 6 multiple of

st th 1 row vector; if a2m+2,2 = e, then the (2m + 2) row vector of A is a — e multiple of 2nd row vector, where 1 < m < ^ — 1. Prom this we have deduced that A depends on only (n — 1) complex numbers with modulus at most 1, for n is even. 98 When n = 2k +1, k € N, we may assume that there are k (—l)'3, (& +1) 1" along the main diagonal line of A, and a(A) = { £ + yjf - ..., £ - yj f -£ }.

Let A' = ^4 — ^/ = (yl')* e J%(C) with entries along the main diagonal of v4.' are

A: (—1 — £)/s, (A;+1) (1 — A)'s and the other entries are complex numbers with modulus at most 1. Since rank(A') = 2, we can apply the same strategy as above to find the structure of A', which will then yield the structure of A.

There are 3 types of invertible 2x2 principal submatrices of A' ( \ / \ ( \ 1-i 1 •i-i a n a

a 1 a _1 _ I V » / \ n / \ l-'l type 1 type 2 type 3

As n —• oo, the determinants of the principal submatrices of type 2 and type 3 approach 0. So we can determine the other row vectors as above by identifying the upper left invertible principal submatrix. From this we arrive at the structure of A! by study the principal submatrix

'l-i a b d 9

— 1 — - a n c e h

1 b c 1 - n J/ i

-1- 1 d e Jf n j

9 h * J

Let the row vectors in the above submatrix be a, /?, 7,5, and 7,5,(, are combinations

99 of a, j3 with combination parameters (s,t), (p, q), (u,v), respectively. It follows that

s(l - £) + ta = b

sa + t(—1 — £) = c (3.46) sd + te = f

sg + th = i,

and

PC1 _ n) + Qa = d

pa + q(-1 -i)= e

pb + qc = f (3.47)

pd + qe = -1 - i

+ qh = j,

and

w(! - k) + vn = 9

ua + v(—\ — -) = h

ub + vc = i (3.48)

ud + ve = j

ug + vh = 1 - i.

If 7i = 3, we only have the first three equations in (3.46). These equations reduce to 2Re(abc)= 2—^ — +^ and 2Re(ade) = — (2 + £ — ^ ^).

As n —> oo,Re(abc) —> 1. So abc = l,t = 0, s = 6; and Re(ade) = —1, then ade = —1 ,p = 0, q = —e.

So if n is odd, the entries of A are determined by the same idea and 1,-1 occurs along the diagonal of A in pair, the last diagonal entry is 1, and if a2m+1,1 = b, then 100 th st = the (2m + l) row vector of A is b multiple of 1 row vector; if a,2m+2,2 then the

(2m + 2)th row vector of A is —e multiple of 2nd row vector, where 1

101 3.6.2 Complex Hermitian Matrices with Entries of Modulus

Let A = [ay] 6 J%,(C), ||ay|| < 1> then SP(A) = Aj — A„, if Ai > A2 >

• • • > An, all the eigenvalues of A. In fact SP(A) = Ai — An = x* Ax — y*Ay —

T e (A o (xx* — yy*))e, where x,y are the associated eigenvectors of Ai and An. Now define A = [ay], ay = cos (—ay) + i sin (—ay) where B = (xx* — yy*) — B* —

(bij), bij = rfj-(cos (ay) + isin (ay)) and ay = ry (cos (/3y) + isin (/3y)), then

T SP(i4) < e (i o B)e = x*Ax - y*Ay < Ai(i) - An(A), and

SP(A) = diibu 4- 2Re ay6y. l

Remark: According to Theorem 2.2.3, when A = [ay] € Jifn, ||ay || = 1, SP(A) <

2 2 ^/2||J4||^ = \/2n, if n is even; and SP(A) < ^2||J4||J. — ^\tr(A)\ < yj2n — if n is odd.

Examples of some matrices A = [ay] G |[ay|| = 1, which attain the maximum spread are included below.

• n = 2 : SP(A) = 2y/2,

1 A

_1 )

102 • n = 3 : SP(A) = ^18-1,A, = i + A2 - i, A3 = J - if

tr(A) = 1 and Re(abc) = —27' / \ 1 a 6

ale

6 c —1 V

• n = 4 : SP(A) = 4y/2, Ai = — A4 = 2\/2, A2 = A3 = 0, if ir(;4) = 0 and

Re(def +bcf) = —Re(abd +ace), 2Re(abd +ace) + Re(abef +acdf +bcde) = 3 ( \ lab c

aid e

b d -1 f

c e f -1

103 3.6.3 Skew Hermitian Matrices with Entries of Modulus No

More than 1

Let A = [ay] € ||ay|| < 1, which implies that A* — —A and iA =

B — [bij] € so by the results we have in the previous section on Hermitian matrices, when SP(A) is maximum and attains the upper bound in Theorem 2.2.3 and rank(A) = 2, the structure of A is as follows,

Case 1: If n is even, £[{1,2,2m + 1,2m + 2}; {1,2,2m + 1,2m + 2}] is the following

matrix / \ 1 a b —ae

a -1 ab e

b ab 1 —abe

—ae e —abe -1

In other words, 1, -1 occurs along the diagonal of B in pairs, and if fe2m+i,i = b,

then the (2m + l)th row vector of B is b multiple of 1st row vector; if &2m+2,2 = e,

then the (2m + 2)ih row vector of B is —e multiple of 2nd row vector, where

1 < m <| — 1. Then we have the structure of B which can be decided by

(n — 1) complex numbers with modulus no more than 1, The same conclusion

then holds for A.

Case 2: If n is odd, entries of B is decided by the same argument and 1,-1 occurs

along the diagonal of B in pairs, the last diagonal entry is 1, and if 62m+i,i = b,

ih 4t then the (2m+ l) row vector of B is b multiple of l row vector; if b2m+2,2 = e,

104 then the (2m + 2)th row vector of B is — e multiple of 2nd row vector, where

1 < m < — 1, and the rule applied in the last row is same as the one in the

3rd row.

Remark: According to Theorem 2.2.3, when A = [a^] € Ua^H = 1, SP(A) <

Y/2||J4||J. = \/2n, if n is even; and SP(A) < y^2||A|||. — ^\tr(A)\2 < yj2n2 — if n is odd.

Note: When n is odd, for the structure we have studied,

4 11 2 Imiabc) = 2 H—- = 2Im(agh) n n2 nA and

2Im(ade) = -(2 + - -\ - n n2 n6

105 Chapter 4

Spectral Inclusions for Normal Matrices

In this chapter, we review some literature on spectral inclusion regions, which concentrates on the relation between a normal matrix and its principal submatrices.

This idea affords another avenue to the maximum spread of a certain matrix.

4.1 New Spectral Inclusions for Normal Matrices

We present some work on spectral inclusion regions for normal matrices [1], and then we find some connections between these spectral inclusion regions and the max imum spread for normal matrices.

Before we come to these conjectures, we define some notation which will be used throughout all this section.

Let A = (ay) be an n x n complex matrix with spectrum cr(A) = {A»}"=1. From this section, we denote by the (n — 1) x (n— 1) principal submatrix of A obtained by deleting the i-th row and column from A and let cr(Ai) = {/Aj}"=i be the spectrum

106 of Ai, and, for 1 < i < n, define

,1/2 ri = ^2 i0*-' and let D(a,r) be the closed disc with radius r € R+ and center a 6 C.

Conjecture 4.1.1 ([1]) For any n x n normal complex matrix A,

n n-1 ff(A) C (J (J Dfoiy.r-i). (4.49) i=l j=1

Recall the Gerschgorin classical disc theorem,

a(A) C |jD(aif ,Pi), where pi = ^ |a

Note that (4.49) uses n(n — 1) discs centered at the eigenvalues of the principal submatrices Ai of A, while (4.50) uses n discs centered at the diagonal entries of A but the related radii in (4.50) is bigger than those in (4.49) since r» < Pi for 1 < i < n.

Conjecture 4.1.2 ([1]) Let A be a singular n x n normal complex matrix. Then there exists i e {1,..., n} such that

min M < n. tt€a(Ai)

It is easy to see that Conjecture 4.1.1 is equivalent to Conjecture 4.1.2 since in the first conjecture we can replace A by A' = A — XI for some A € cr{A), then apply

A' in the second conjecture.

n Let elt...,e n be the standard basis in H := C and denote by Pi the orthogonal

1 projection on q , so that P{H = ef and Pi H \- Pn = {n - 1)1. One has

Ai = PiAPilp^ and ||(/ — Pi)APi\\p = r^ for 1 < i < n. 107 Clearly, if A is a rank-one operator then ||^4||F = IL-<4||> where ||-|| denotes the operator norm. Here

n = ||(/ - POAPiWf = ||(7 -Pi)APi\\ for 1 < i < n.

The following is another related conjecture from [1],

Conjecture 4.1.3 ([1]) Let {P»}"=1 be a family of rank (n — 1) projections in n- dimensional Hilbert space H such that (7 — Pj)-L(7 — Pj) for 1 < i ^ j < n. For any singular normal operator A € L(Ti) one has

mm min \n\ - 11(7 - Pi)APi\ | < 0, l

Evidently, Conjectures 4.1.1-4.1.3 are true for diagonal matrices. When n = 2, these conjectures become precisely Gerschgorin's Disc Theorem and hence are true in this case. When n = 3, we have the following theorem.

Theorem 4.1.4 ([1]) Conjecture ^.l.l-J^.1.3 are true for n = 3.

To prove it, we need Banach Spaces and to construct a surjective linear operator with surjective constant M (see [1]).

There is a better covering conclusion when A is a Hermitian matrix.

Theorem 4.1.5 ([1]) If A is ann x n Hermitian matrix then

+ so that a(A) C (J"=1 (J7=i ~ 108 Theorem 4.1.5 will follow the lemmas below.

n Lemma 4.1.6 ([1]) Let n > 2 and v = (vi,..., vny € C be such that at least two

of its components are non-zero. Then

1 min 7==t; M:—< ^

Lemma 4.1.7 ([1]) If A is an n x n singular Hermitian matrix then there exists

1 < i < n such that Ti min |^| < H€

Proof. Evidently, it is true if there is at least one (n — 1) x (n — 1) singular principal

T n submatrices of A. Let v = (vi,...,v n) 6 C \ {0} and Av = 0. If all but one

th component of v are zero, assume that vn ^ 0 then the n column of A is a zero vector. Hence there axe at least n — 1 singular principal submatrices of A\,..., An and then the lemma is obvious in this situation. Now we assume that v has at least two non-zero components. By Lemma 4.1.6, we have vn ^ 0 and

i?i i j < . (4.51) l/2 y/n — 1 feSM2 Let

,(n) =_ (^1) • • • ) ^n-l)^ € Cltpin—l_1 ^ {Q}) so that ||i /n^ [ 12 = 1. Prom Av = 0 we have

A fn} ^n(Olri) • • • i &n—In) nV ~ 1 1/2 • m2

109 Since both An and A are Hermitian it follows from the last identity and (4.51) that

m (n) in M = inf Pnx||2 < ||A,v ||2 ti€o(An) ||i||2=l 1/2 1/2 < n — 1 n — 1 y/n — l' which proves the lemma. •

Theorem 4.1.5 now follows by applying Lemma 4.1.7 to the n x n singular Her mitian matrices A — A I,A € cr(A).

Thus, Borcea describes the covering of eigenvalues for normal matrices, order n> 3, by discs generated from all its principal submatrices, order (n — 1). Applying

Conjecture 4.1.1 (or Conjecture 4.1.2) to normal matrices, order n > 3, whose spreads are attained the maximum value, we will find the eigenvalues of these normal matrices are covered by the related n(n— 1) circles, since for these matrices, the corresponding rank is 2, which means they are singular.

Lemma 4.1.8 For a € [—1, l),n € N, we have round(^|) = round(^3) otherwise, round{—^) = round(^) - 1.

Lemma 4.1.8 shows that if A € Sn-{a, 1} and A has the structure in (3.40), then its principal submatrices of order (n — 1), the maximum spread is attained for A\ or An.

Meanwhile, in this case, A and its principal submatrices are all Hermitian, by the eigenvalue interlacing theorem and Theorem 4.1.5, we have the following conjecture.

Conjecture 4.1.9 A € ^-{a, 1} and A has the structure in (3-40), then

SP(A) < max{SP(Ai), SP(An)} + 2. 110 For the matrix A € 55n[—1,1], when the spread is maximum, then A is of the form in Theorem 3.5.7. Then we have the following conjecture.

Conjecture 4.1.10 For the matrix A € SSnf— 1, l],max{SP(A)} attains, then A has the structure presented in Theorem, 3.5.1, and

SP(A) < max + 2^=5,SP(A r+1) + 2^=,SP(A r+s+1) + 2^=} , where r, s, t are the corresponding order of the block zero matrix, and r + s + t = n.

There are many works focusing on the relation between the normal matrix and its principal submatrices. We introduce them in the next section.

Ill 4.2 Imbedding Conditions for Normal Matrices

Given a complex n x n matrix A and (n — k) x (n — A;) matrix B, we say 5 is imbeddable in A or, equivalently, B is a compression of ^4 if B is a principal submatrix of U*AU for some nxn unitary matrix U.

The question, when an (n — k) x (n — k) normal matrix B can be imbedded in an nxn normal matrix A, was studied for the first time 50 years ago by Ky Fan and Gordon Pall [5], who gave a complete answer for k = 1. In 1984, Carlson and

Marques de Sa [3] proved a theorem on the same problem. Later, in 1998, Ikramov and Eisner [12] studied the slightly different question of finding conditions under which an (n — k) x (n — k) normal matrix B can be-nontrivially-dilated to an n x n normal matrix A, with special attention to the k = 1 and k = 2 cases.

Since the spread of a normal matrix is invariant under unitary similarity, we want to build up a connection among Borcea's conjectures, spread, and imbedding conditions for normal matrices.

The following theorems gives necessary and sufficient conditions on imbedding for normal matrices.

Theorem 4.2.1 ([5]) Let A be annxn normal matrix with eigenvalues c*i,..., a„,

B an (n — 1) x (n — 1) normal matrix with eigenvalues Pi,... ,Pn-i- Renumber the eigenvalues so that ctj = Pj-\, j — q + 1,..., n and oti,...,aq are each dis tinct from Pq—i• Then B is imbeddable in A if and only if the 2q — 1 points

<*i,..., aq, ,..., /?g_x are collinear and the P's separate the a's on that line.

112 In the Hermitian case, using Cauchy's interlacing theorem, only the n — 1 case is needed for induction in the general n—k case, which means we just insert intermediate sequences of eigenvalues for a chain of matrices. But we can't apply the same strategy in the normal case, as shown by the following example in [5]. Take

A = diag(0,1, i,1 + i), and B = ^diag(5 4- 8i, 5 4- 2i).

Let 1 2

1 -2 V = VIo 2 -1

V / then V*V = I2 and V*AV = B so B is imbeddable in A and by Theorem 4.2.1, any normal matrix C of order 3 imbeddable in A must have two of 0, l,i,1 4- i as characteristic roots and its third root on the segment joining the remaining two. But by Theorem 4.2.1, B can not be imbedded in any such matrix C.

Carlson and Marques [3] proved a more general result on the same problem.

Theorem 4.2.2 Let A be an nxn normal matrix with nonzero eigenvalues <*1,...,a n, satisfying 7 4- 7r > argot 1 > • • • > argan > 7 for some 7 > 0. Let B be an

(n — k) x (n — k) normal principal submatrix of A. The eigenvalues 0i,... ,/?„-* of B may be ordered so that 7 4- n > argfc > • • • > arg(3n_k > 7 and

argccj > argfy > argaj+k, j = 1,..., n - k.

Joao Filipe Queiro and Antonio Leal Duarte [24] applied the idea of using the lexicographic order in C to mimic the Hermitian case on this problem. 113 For real 9, let

H = {a + ib : a > 0, or a = 0 and b > 0}.

So the lexicographic order is

The condition a >g 0 means that, when we sweep the plane with parallel lines orthogonal to the argd direction, intersecting that direction before going to the arg(9 + TT) one, we find a before we find /? (in case of a tie, a is found to the right of (3). The numbering of elements in decreasing sequences of course depends on

Now let 0 e R be arbitrary, take an nxnnormal matrix A, and let alt... ,an be the eigenvalues of A, ordered so that ai >$ • • • >e an, vlt... ,vn be corresponding orthonormal eigenvectors of A. For j = 1,... ,n, denote by Ej and E'j the subspaces spanned by vi,..., Vj and Vj,...,v n, respectively. Then in similar circumstances max and min used in the

Theorem 4.2.3 For j = 1,..., n we have

a,- = min x*Ax = max x* Ax. x€Ej,\\x\\=l *€£5,11*11=1

They also proved that

a, = max min x* Ax = min max x*Ax, dimE—j, i€£,||z||=l dimE=n-j+1, I€J5,||I||=1 and the interlacing theorem for normal matrices. 114 Theorem 4.2.4 Let 6 be arbitrary and A be an nxn normal matrix with eigenvalues

c*i >0 • • • >0 an. If B is a principal (n — k) x (n — k) normal submatrix of A with eigenvalues Pi >g - • • >$ Pn-k, we have

&j ^0 Pj ^0 &j+ki j == 1, • . . j 71 k.

115 4.3 The Spread of a Normal Matrix and Its Principal Sub-

matrices

There are many papers that focus on the issue of relating the eigenvalue spread of a matrix and its principal submatrices.

Thompson has a series of studies [25, 26, 28] on this problem for a normal matrix and its principal submatrices. There are many other studies inspired by his research, like Johnson and Robinson [14, 15], and Nylen and Tam [22]. We mainly introduce

Thompson's results.

Theorem 4.3.1 ([26]) Let A be a Hermitian matrix with eigenvalues Ai > • • • > \n,

and Ai the principal submatrix of A obtained by deleting row i and column i,i = l,...,n. Then n

Y^SP(Ai)>(n-2)SP(A). i=l

The main tool used to approve it are the interlacing theorem (since the principal submatrices of the Hermitian matrix are all Hermitian), and adjugate matrix.

Then Thompson asserted a quadratic spread conjecture by abundant computer generated numerical evidence. He proved it is true for n = 3.

Conjecture 4.3.2 ([26]) Let A be a Hermitian matrix and Ai are the principal submatrices. Then

J2SP(Ai)2>(n-2)SP(A)2. i=l 116 Later Nylen and Tam [22] showed that this conjecture is false for n x ra Hermitian matrices when n > 4. However, it is true if the coefficient n — 2 is replaced by some smaller value which is depended on n.

Thompson [27] also studied the generality of quadratic spread conjecture for Her mitian matrices to a quadratic spread conjecture for Hermitian matrix pencils.

Define a Hermitian matrix pencil AH — K, where A is an indeterminate, and

H, K are n x n Hermitian matrices, and H is positive definite. Let the roots of the pencil AH — K be Ai,..., An. Then there exists a unitary matrix V such that

2 2 l 2 u V*H~^ KH-^ V = diag(Xi,Xn). Set T = H ' V and let T' = [^], and take

/(A) = det(XH - K) = detH(A — Aj) • • • (A — An), /4(A) = det(\Hi - K - i) as the pencil characteristic polynomial and the characteristic polynomial of the subpencil det(XHi — Ki), respectively. Then (see [27])

j=1 J

Thus fW hW A-Ai

= & (4.52)

fW . /n(A) . A—An where the matrix & = [|£i,j|2] = [rij] is nonnegative but not necessarily doubly stochastic.

Define SP(f(A)) of a polynomial /(A) with real roots to be the difference of its largest root and the smallest one. Then he has the following quadratic spread inequality theorem. 117 Theorem 4.3.3 ([27]) Let ST be an n x n matrix, n > 3, with nonnegative entries

th and no zero rows or columns. Let /(A) be an n degree polynomial with roots Ai,..., An

all real, and use and f to define polynomials /i(A),..., /„(A) as in (4-52). Then

a positive constant c not dependent on the roots of /(A) exists such that

X>f(/.W)]2 a c[sWM]2. t=l

A similar theorem is derived from a Hermitian matrix pencil when define the spread of the pencil AH — K as the separation between the largest and smallest of the roots of the pencil.

Theorem 4.3.4 ([27]) Let H and K by nxn Hermitian matrix with H positive def inite and n > 3. Then a positive constant c = c(H) depending only on the eigenvalues of H and not depending on K, exists such that

n Y^[SP(XHi - Ki)f > c[SP(\H -K)\\ i—l

118 Chapter 5

Conclusion

In this chapter, we review the entire thesis and summarize our work on the max imum spread of certain normal matrices, and develop a plan for future research.

5.1 Summary

Research on the spread of a matrix started in the 1950's, and there is much literature on upper and lower bounds on the spread of some general normal matrices and simple graphs [6,9,13,17 ~ 19,32], By Mirsky's Theorem, Theorem 2.2.3, we began a study on the maximum spread over a certain class of normal matrix whose entries are in an interval or a circle.

For a real normal matrix, if A € Sn[a, 1], a € [—1,1) and rank(A) = 2, and if

SP(A) is maximum, then A must be of the form shown in Theorem 3.4-1; we presented the maximum spread and the corresponding structure of matrix in Conjecture 3-4-3

, when A £ £n[a, b]. For a real skew symmetric matrix, when A = [a^] € 55 n[—1,1], and rank {A) = 2, with \a,ij\ <1, 1 < i, j < n, we have Theorem, 3.5.5 which presents 119 sufficient and necessary conditions of the maximum spread and the related structure

of the matrix.

For a complex normal matrix, when A = [a<3] € J%(C), 1111 < 1, if SP(A)

attains the upper bound in Theorem 2.2.3 and rank(A) = 2, then 1, -1 occurs

consecutively along the diagonal line of the matrix A, while the other entries are

determined by (n — 1) complex numbers with modulus no more than 1. Moreover,

||ajj|| = 1 if and only if the spread of A, SP(A) is maximum. A similar conclusion can be applied in the class of A = [ay] E ||ay|| < 1, since A* = —A and iA = B = [bij] € JC

We have not shown more detail about the case of higher rank of A, a normal matrix

with entries in some specifical range although we know the case of real symmetrical

Mirsky matrix. But we have many numerical results shows rank(A) might be 2 when the maximum spread of this normal matrix attains. And it is more complex if we try to describe the pattern of these matrices with higher rank since there are many combinations related to the extreme (modulus) value of the entries of A in the specific range.

There are many papers that focus on the relation between a normal matrix and its principal submatrices [1,3,5,14,15,22,24 ~ 27]. This array of research inspires us to continue to study the properties of spread of a normal matrix (with the entries in an interval or a circle) in terms of its principal submatrices. Finally, we presented

Conjectures 4-1-9, 4-1-10-

120 5.2 Future Considerations

In the future, I would like to prove Conjecture 3.4-3 and explore the properties of spectral inclusion for a normal matrix, and make more accurate upper bounds on the

spread of a normal matrix in term of the spreads of its principal submatrices.

I would like to investigate the spread property of matrix power, defined as AK =

n*i A, especially, A0 = I. As we know, calculating high powers of matrices can

be very time-consuming, but the complexity of the calculation can be dramatically decreased by using Cayley-Hamilton theorem, which takes advantage of an identity found using the matrices' characteristic polynomial and gives a much more effective equation for Ak. There is not that much literature focusing on the spread of powers of a normal matrix.

Complex symmetric matrices are not always normal. Since complex symmetry is a purely algebraic property, then it has no effect on the spectrum of the matrix. It means, for any given set of n numbers,

Ai, A2,.. •, An € C, (5.53) there exists a complex symmetric n x n matrix A whose eigenvalues are just the prescribed numbers(5.53); see, e.g., [11]. Prom Lemma 2.2.2, we want to concentrate on this topic of maximum spread for complex symmetric matrices and the associated matrices structure.

The spectral gap, difference between the maximum and minimum value of the

121 spread, has been of interest to others for some time. Since the characteristic polyno mial is a continuous function of the entries of the matrix, then the spread is continuous and then attains the lower bound and upper bound in a closed set. We would like to investigate this research further.

122 Chapter 6

Appendix

6.1 The Mat lab Programs

The 1st program of Matlab aims to look for the maximum spread and the related the structure of the matrices which entries in the interval [a, 1], and compare them with the parameters in Theorem 2.2.3.

function sprmax (nk, ak)

n = nk] a = ak; b = —ak; (transfer parameters)

ifa = 0ora = —1 then generate one set values for the number of the block, s0 and the related SP in Theorem 2.2.3 and set the initial value of the loop 123 ka = round(n/(3 + a)); flaga = 0; sa = sqrt((a + 3) * (a — 1) * ka2 4- 2 * n * (1 — a) * ka 4- n2); otherwise generate two sets values ka = round(n/(3 + a)); kb = round(n/(3 + &));

2 2 sa — sqrt((a + 3) * (a — 1) * ka + 2 * n * (1 — a) * fca + n ); Jlaga = 0; sfc = sqrt((b + 3) * (6 — 1) * /c62 + 2 * n* (1 — b)* kb + n2); /Zaffc = 0; process the loop of generating the random matrices if o 7^ 0 and a ^ — 1 then generate 3 x 106 random real matrices Aa with entries a, 1 and Ab with entries

—a, 1

Aa = rand(n); Ab = rand(n)\ for i = 1 : n for j = i : n if Aa(i,j) < 0.5

Aa(i,j) = a;Aa(j,i) = a; else

Aa(i,j) = l;Aa(j,i) = 1;

if Ab(i,j) < 0.5

Ab(i,j) = 6; Ab{j,i) = 6; else

Ab(i,j) = l-,Ab(J,i) = 1;

124 calculate the spread values spra,sprb of matrices Aa, Ab if spra > sa

flaga = 1; Aa, ka if sprf, > sb

flagb = 1; Ab, kb

when the loop is stop check the value of flag. if flaga = 0 disp(['The default matrix, positive a.'])

if flagb = 0 disp(['The default matrix, negative a.'])

if a = 0 or a = —1 then generate 3 x 106 random real matrices Aa with entries a, 1

Aa = rand(n)-, for i = 1 : n for j —i :n if Aa(i,j) < 0.5

Aa(i,j) = a\Aa(j,i) = a; else

Aa(i,j) = 1; Aa(j,i) = 1;

125 calculate the spread values spra, spri, of matrices Aa, Ab if spra > sa

flaga = 1; Aa, ka

when the loop is stop check the value of flag. if flaga = 0 disp(['The default matrix, positive a.']) output the data to the file.

The 2nd program of Matlab aims to look for the matrices which entries in the interval

[a, 1] and satisfy the property of Mirsky's Theorem.

function Mirsky (nk, ak)

n = nk] a = ak\ b = — ak] transfer the parameters process 3 x 106 loops of generating the random matrices Aa with entries a, 1 and Ab with entries —a, 1

Aa — rand(n);Ab = rand(n); in every loop assign the entries of the matrices 126 for i = 1 : n for j = i : n if Aa{i,j) < 0.5

Aa(i,j) = a;Aa(j,i) = a; otherwise Aa(i,j) = 1; Aa(j,i) = 1;

if Ab(i,j) < 0.5

Ab(i,j) = b;Ab(j,i) = b; otherwise Ab(i,j) = l]Ab(j,i) = 1; calculate the eigenvalues of Aa, Ab and output to la, lb compare the other eigenvalues with the average value of the 1st and nth in la, lb; if all the remaining eigenvalues are equal to the mean, then output the corresponding matrix.

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131