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Majorization Via Generalized Hessenberg Matrices Q CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Linear Algebra and its Applications 292 (1999) 127±138 Majorization via generalized Hessenberg matrices q Suk-Geun Hwang * Department of Mathematics Education, Kyungpook National University, Taegu 702-701, South Korea Received 11 November 1997; accepted 12 January 1999 Submitted by R.A. Brualdi Dedicated to Professor Richard A. Brualdi on the occasion of his 60th birthday Abstract In this paper we give a method of constructing generalized Hessenberg matrices from those of smaller orders, with a simple combinatorial justi®cation. Making use of this construction, we also prove that, for real n-vectors x and y whose components are ar- ranged in nonincreasing order, x is majorized by y if and only if there exists a doubly stochastic matrix A of order n with x Ay whose support is permutation similar to a direct sum of generalized Hessenberg matrices. Ó 1999 Elsevier Science Inc. All rights reserved. Keywords: Hessenberg matrix; Generalized Hessenberg matrix; Majorization 1. Introduction The following n  n (0,1)-matrix is called the lower Hessenberg matrix or simply a Hessenberg matrix of order n, q This paper was supported by Non Directed Research fund, Korea Research Foundation in 1996. * E-mail: [email protected]. 0024-3795/99/$ ± see front matter Ó 1999 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 4 - 3 7 9 5 ( 9 9 ) 0 0 0 2 3 - 3 128 S.-G. Hwang / Linear Algebra and its Applications 292 (1999) 127±138 2 3 1 1 0 0 Á Á Á 0 0 0 6 1 1 1 0 Á Á Á 0 0 0 7 6 7 6 1 1 1 1 Á Á Á 0 0 0 7 6 7 6 . 7 6 1 1 1 1 . 0 0 0 7 6 7 H 6 . 7: n 6 . 7 6 7 6 1 1 1 1 Á Á Á 1 1 0 7 6 7 6 1 1 1 1 Á Á Á 1 1 1 7 4 1 1 1 1 Á Á Á 1 1 1 5 In [4], Brualdi and Shader generalized the notion of Hessenberg matrices as follows. Let A a1; a2; . ; an be an n  n (0,1)-matrix and let 0 A ak A T ; ek 1 where ek denotes the kth column of the identity matrix In of order n. The n 1  n 1 matrix A0 is called the expansion of A at column k. An ex- pansion of A at a row is de®ned similarly. A square (0,1)-matrix A is called a generalized Hessenberg matrix if there is a sequence of matrices A0; A1; . ; At such that A0 I1; At A and Ak is an expansion of Ak1 for k 1; 2; . ; t. An n  n matrix is called fully indecomposable if it does not contain an s  n s zero submatrix. By de®nition, we easily see that a generalized Hessenberg matrix is fully indecomposable. Let A; B be square (0,1)-matrices of orders k and m, respectively, and let r; s be integers such that 1 6 r 6 k; 1 6 s 6 m. Let A r } B s denote the matrix obtained from A È B by replacing each of its rows r and k s by the sum of those two rows, i.e., let A r } B s In Jr;ks A È B; where Jr;ks denotes the matrix obtained from In by interchanging rows r and k s and then replacing each of its diagonal entries by zero. If A is a square T (0,1)-matrix, then the expansion of A equals either of the matrices A k } I1 1 T T or A k } I1 1 for some k 1; 2; . ; n. Let Xn denotes the polytope con- sisting of all n  n doubly stochastic matrices. Let D dij be an n  n (0,1)- matrix. D is said to have total support if for every pair of indices k; l with dkl 1 there is a permutation matrix P pij with pkl 1 such that P 6 D, where and in the sequel, for two real matrices A; B of the same size, A 6 B means that every entry of A is less than or equal to the corresponding entry of B. Let X D fA 2 Xn j A 6 Dg: Then X D is a face of Xn and every face of Xn is obtained in this fashion [2]. For a polytope } in a ®nite dimensional Euclidean space, let dim } denote the dimension of }. While a generalized Hessenberg matrix has many properties in S.-G. Hwang / Linear Algebra and its Applications 292 (1999) 127±138 129 common with a Hessenberg matrix (see [1,4] for example), some property of a generalized Hessenberg matrix is better or simpler than that of a Hessenberg matrix. For example, dim X G 6 n n 1=2 for any generalized Hessenberg matrix G of order n, while dim X Hn n n 1=2. Let Rn denote the set of all real column vectors with n components. For T n T a a1; a2; . ; an 2 R , let a # a1; a2; . ; an denote the vector ob- tained from a by rearranging the components in nonincreasing order. For x T T n x1; x2; . ; xn ; y y1; y2; . ; yn 2 R ; x is said to be majorized by y, written x y, if Xk Xk xi 6 yi i1 i1 for all k 1; 2; . ; n, where the equality holds if k n. It is a well known classical theorem of Hardy et al. [7, p. 47] that, for x; y 2 Rn; x y if and only n if x Sy for some S 2 Xn. For x; y 2 R with x y, let Xn x y fS 2 Xn j x Syg: Then Xn x y is a subpolytope of the polytope Xn. This polytope is known to contain doubly stochastic matrices of very special types [9]. In [1], Brualdi investigated the structure of the polytope Xn x y along with some other properties of it. It is known that the set Xn x y contains an orthostochastic matrix [9], a uniformly tapered matrix [9], and a positive semide®nite matrix [5]. For two square matrices A and B of the same order, A is said to be per- mutation equivalent to B if there exist permutation matrices P and Q such that PAQ B. A is said to be permutation similar to B if there exists a permutation T matrix P such that P AP B. For a real matrix A aij, the matrix S sij of the same size as A, de®ned by 1 if aij 6 0; sij 0 if aij 0 is called the support of A and is denoted by r A in the sequel. Recently, Brualdi and Hwang [3] showed that Xn x y contains a matrix whose support is permutation equivalent to a direct sum of Hessenberg matrices, which proves that x is expressible as a convex combination of n2 n 2=2 or less per- mutations of y. In this paper we show, with a simple combinatorial proof, that if A; B are generalized Hessenberg matrices of orders k and m respectively, then A r } B s is a generalized Hessenberg matrix for any r; s with 1 6 r 6 k; 1 6 s 6 m. This gives us another way of constructing generalized Hessenberg matrices from those of smaller orders. As an application of this construction we also show that Xn x # y # contains a matrix whose support is permutation similar to a direct sum of generalized Hessenberg matrices. This 130 S.-G. Hwang / Linear Algebra and its Applications 292 (1999) 127±138 result can still prove that x is expressible as a convex combination of n2 n 2=2 or less permutations of y. 2. Generalized Hessenberg matrices In the following we give a graph theoretical description of generalized Hessenberg matrices. In the sequel, by a bipartite graph we mean one with no multiple edges. First let C be a bipartite graph with bipartition of vertices U; W . Suppose that U fp1; p2; . ; pmg and W fq1; q2; . ; qng, and let A aij be the m  n (0,1)-matrix with the property that aij 1 if there is an edge connecting pi and qj, and aij 0 otherwise. The matrix A is called the reduced adjacency matrix of C, and C is called the associated bipartite graph of the matrix A. In view of the above de®nition, we call the vertices pi's and qj's the row vertices and the column vertices of C, respectively. For a vertex v of C, let NC v denote the set of all vertices which are adjacent to v in C. In the literature, the set NC v is called the neighborhood of v and each of the elements of NC v is called a neighbor of v in C. In this paper, we say that C is of order n if jUj jW j n, where, for a set S, jSj denotes the number of elements of S. Suppose that C is of order n. We extend C to a bipartite graph of order n 1 as follows. Step 1. Pick a vertex v of C. 0 0 Step 2. Introduce a new vertex v and connect every vertex in NC v to v with a new edge.
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