UNIVERSITY OF OSLO Department of Informatics

A note on nonnegative diagonally dominant matrices

Geir Dahl

Report 269, ISBN 82-7368-211-0

April 1999

A note on nonnegative diagonally dominant matrices

∗

Geir Dahl April 1999

e make some observations concerning the set C of real nonnegative,

W n

diagonally dominant matrices of order . This set is a

symmetric and n

convex cone and we determine its extreme rays. From this we derive

dierent results, e.g., that the rank and the kernel of each matrix

A ∈Cn is

, and may b e found explicitly. y a certain supp ort graph of determined b A

ver, the set of doubly sto chastic matrices in C is studied.

Moreo n

Keywords: Diagonal ly dominant matrices, convex cones, graphs and ma-

trices.

1 An observation

e recall that a real matrix of order is called diagonal ly dominant if

W P A n

| |≥ | | for . If all these inequalities are strict, is

ai,i j=6 i ai,j i =1,...,n A

strictly diagonal ly dominant. These matrices arise in many applications as e.g.,

discretization of partial dierential equations [14] and cubic spline interp ola-

[10], and a typical problem is to solve a linear system where

tion Ax = b

strictly diagonally dominant, see also [13]. Strict diagonal dominance

A is

is a criterion which is easy to check for nonsingularity, and this is imp ortant

for the estimation of eigenvalues confer Ger²chgorin disks, see e.g. [7]. For

more ab out diagonally dominant matrices, see [7] or [13]. A matrix is called

nonnegative positive if all its elements are nonnegative p ositive.

n,n

D ⊂ denote the set of all matrices of order that are nonnegative

Let n IR n

and diagonally dominant. The set of symmetric matrices in y Dn is denoted b

∗

D . Both these sets are p ointed p olyhedral convex cones in the vector space n

n,n

as wehave IR of real matrices of order n n,n ≤ ≤ Dn={A∈IR : ai,j ≥ 0 for 1 i, j n;

P

≥ for } 1 ai,i j=6 i ai,j i =1,...,n ,

D∗ { ∈D for ≤ ≤ } n = A n : ai,j = aj,i 1 i, j n .

University of Oslo, Dept. of Mathematics and Dept. of Informatics, P.O.Box 1080,

Blindern, 0316 Oslo, Norway Email:geird@i.uio.no 1

n,n

t matrices in is a nonconvex

Note that the set of diagonally dominan IR

The interior of D consists of the p ositive and strictly diagonally dominant cone. n

∗

, the relativeinterior of D consists of the symmetric, p ositive

matrices. Similarly n

and strictly diagonally dominant matrices.

We mention an interesting result from [9] that is relevantto this note. It

∗

as shown that if ∈D , then is completely p ositive. This means that w A n A A

T

for some nonnegative × matrix. We return

can b e factored as A = BB n m

to this result in connection with Theorem 3 b elow.

{ } b e a set of ≥ vectors in a vector space over the

Let S = v1,...,vk k 1 V

reals. The nitely generated convex cone

Xk { ≥ } cone(S)= λj vj : λ1,...,λk 0

j=1

spanned by . If the vectors are linearly indep endent,

is said to b e S v1,...,vk

is called a simplex cone. We need a simple result on such cones.

cone(S)

Let cone be the convex cone spanned by { }⊂ .

Lemma 1 (S) S = v1,...,vk V

Then cone a simplex cone if and only if each point in cone may be

(S) is (S)

written uniquely as a conical i.e., nonnegative linear combination of the vectors

v1,...,vk.

If are linearly indep endent, then the representation is clearly

Pro of. v1,...,vk

Conversely, assume that the uniqueness of such representations hold and unique.P

k 0

. Cho ose nonnegativenumb ers and for that j=1 µj vj = 0 P λj Pλj j =1P,...,k

0 0

h that − for each . Then − . suc µPj = λj λPj j 0 = j µjvj = j λj vj j λj vj

0 0

so by assumption and for all .

Therefore j λjvj = j λjvj λj = λj µj =0 j

are linearly indep endent. ws that

This sho v1,...,vk

e call the unique representation of a p oint in a given simplex cone the

W v

onical representation of . c v

n

denote the th unit vector in and dene the following matrices of

Let ei i IR : order n

i T

for ; i ∆ = eiei i =1,...,n

i,j T

for ≤ ≤ ; 2 ii ∆ =(ei+ej)(ei + ej ) 1 i

¯ i,j T

for 6 . iii ∆ = ei(ei + ej ) i = j

i

These are all has a single one which is in p osition . The (0, 1)-matrices. ∆ (i, i)

i,j

are in p ositions and . Finally, four ones in the matrix ∆ (i, i), (i, j), (j, i) (j, j)

¯ i,j

wo ones, in p ositions and . Let S b e the set of matrices in ∆ has t (i, i) (i, j) n

∗

S b e the set of matrices in 2i and ii. Note that

2i and iii, and let n

all these matrices are nonnegative, diagonally dominant and have rank one.

∗

ver, the matrices in S are symmetric and p ositive semidenite. Moreo n

∗ ∗

D cone S and D cone S . Moreover, both D and Prop osition 2 n = ( n) n = ( n) n

∗

D are simplex cones.

n 2

∗

Let ∈ cone S so there are nonnegativenumb ers for ≤ and

Pro of. A ( n) λi i n

≤ ≤ such that λi,j for 1 i

Xn X i i,j (∗) A = λi∆ + λi,j ∆ .

i=1 i

rom this it follows that is symmetric as a linear combination of symmetric

F A

e. Moreover, ∗ gives for ≤ ≤ matrices and nonnegativ ( ) ai,j = λi,j 1 i

i,j

has a nonzero in p osition . Moreover, due to the as only the matrix ∆ (i, j) P

i,j

we also get from ∗ that

Pstructure of the matricesP ∆ P ( ) ai,i = λi + j

− . From this we conclude that i

∗

e and diagonally dominant and therefore ∈D . Conversely, A is nonnegativ A n

∗ ∗ ∗

h ∈D may b e written in the form ∗ so we conclude that D cone S . eac A n ( ) n = ( n)

∗

ver, we see that each ∈D has a unique representation as a conical Moreo A n

i i,j ∗

bination of the matrices and . So, according to Lemma 1, D is a

com ∆ ∆ n

The pro of of the results for D is similar.

simplex cone. n

Related results on generators for certain cones are found in [1]. They study

t convex cones asso ciated with diagonally dominant matrices, e.g., the

dieren P

satisfying ≥ | | so the only

complex or real matrices of order n ai,i j=6 i ai,j

nonnegativity requirements are on the diagonal elements. Note that the set of

these matrices is a convex cone, but not a simplex cone.

We hereafter concentrate our study on the symmetric diagonally dominant

∗ D .

matrices, i.e., the set n

From the previous pro of we see that the conical representation of a symmet-

e and diagonally dominant matrix is simply ric, nonnegativ A

Xn X X i i,j

A = (ai,i − ai,j )∆ + ai,j ∆ . 3 i=1 j=6 i i

∗

e dene the support graph of a matrix ∈D as the graph

W A n GA =(V,EA)

no de set { } and edges i a lo op when

Pwith V = v1,...,vn [vi,vi] ai,i >

and ii when for ≤ ≤ .

j=6 i ai,j for i =1,...,n [vi,vj] ai,j > 0 1 i

us, edges of corresp ond to the p ositive co ecients in the conical repre-

Th GA

tation of . This graph will b e used b elow. sen A

E

is some nite set, ⊆ and ∈ we use the notation

P When E S E x IR x(S):=

We also let ∅ .

e∈Sxe. x( ):=0

2 Some consequences

Wenow lo ok at some consequences of our prop osition.

Dimension and faces. An immediate consequence of Prop osition 2 is that

2 ∗

D so it is full-dimensional and D − . dim( n)=n dim( n)=n(n 1)/2

∗

kernel. In order to study the kernel of matrices in D we need some The n

∗

Consider a matrix ∈D . Let ⊆ b e the con-

graph notations. A n C1,...,Ct V

nected comp onents of the supp ort graph Wemay assume after reordering

GA . 3

≤ ≤ the subgraph is bipartite and without any lo op. Here

that for 1 j p GA[Cj]

means that no comp onent is bipartite and without lo ops. 0 ≤ p ≤ t and p =0

+ − + −

or each ≤ , let and b e the two color classes of . Thus, , F j p Cj Cj Cj Cj Cj

+

and each edge of joins a no de in andanodein is a partition of Cj GA[Cj ] Cj −

Cj n Cj +

Let ∈ beavector whose supp ort is , and if ∈ Cj. z IR Cj zi =1 vi Cj

Cj −

− if ∈ . If the color classes change role, we obtain the nega- and zi = 1 vi Cj

Cj

eof , but this ambiguity will not matter b elow. Note that we allow the

tiv z

t to b e trivial, i.e., with a single no de but no lo op, and then comp onen Cj vi Cj

z = ei .

ehave the following result on the kernel Ker { ∈ With this notation w (A)= x

n ∗

} of a matrix ∈D . IR : Ax = 0 A n

∗

∈D . Then rank − and Let Theorem 3 A n (A)=n p

C1 Cp

span { } Ker(A)= ( z ,...,z ). P P

n i i,j

Consider the conical representation Pro of. P A = j=1 λi∆ + i

n

− ≥ and ≥ . Let ∈ . From the where λi = ai,i j=6 i ai,j 0 λi,j = ai,j 0 x IR

i i,j

and we obtain the following identities simple structure of the matrices ∆ ∆ P P n i Ax = i=1 λixiei + i

P P 4 T n 2 2

ii x Ax = i=1 λixi + i

Moreover, in 4 it suces to sum over those which ∈ i.e.,

i for [vi,vi] EA

and those for which ∈ i.e., . Note that λi > 0  i 0

T

is p ositive semidenite. x Ax ≥ 0 so A

T

w ∈ Ker . Then and therefore . Thus, from

Let no x (A) Ax = 0 x Ax =0

4ii we see that

whenever ∈ , and

a xi =0 [vi,vi] EA

− whenever ∈ and 6 .

b xi = xj [vi,vj] EA i = j

rom a and b it easily follows that for each no de that lies in

F xi =0 vi

comp onent of  which contains an odd cycle or a lo op. Consider a

a Cj GA

t where ≤ ≤ so is a bipartite comp onent with no

comp onen Cj 1 j p GA[Cj ]

lo op. Then it follows from b that, for some real number for each α, xi=α

+ −

∈ and − for each ∈ . Thus, the restriction of to the vi Cj xi = α vi Cj x

Cj

lies in span . This holds for every ≤ and we conclude that no des in Cj (z ) j p

C1 Cp { } . x ∈ span( z ,...,z )

C1 Cp

∈ span { } . Then whenever versely, assume that

Con x ( z ,...,z ) xi =0

ts . Moreover, for each edge ∈

vi lies in one of the comp onen Cp+1,...,Ct [vi,vj]

b elongs to one of the comp onents we have − . If

EA that C1,...,Cp xi = xj

a trivial comp onent with no lo op, then

Cj = {vi} is λi = λk,i = λi,j =0

≤ and ≤ , so b oth the th row and the th column of

for 1 k

ector. From these observations and 4i it follows that

are the zero v Ax = 0

∈ Ker . This proves the description of the kernel. Finally, we note

so x (A)

that all the vectors spanning the kernel are nonzero and have disjoint supp orts, 4

they are linearly indep endent. Therefore, the kernel has dimension and

so p − .

rank(A)=n p

From this result we see the interesting fact that the kernel and the rank

a symmetric, nonnegative diagonally dominant matrix dep ends only on

of A

the supp ort graph. In other words, the kernel and the rank are determined by

which co ecients in the conical representation 3 that are p ositive; otherwise

the magnitudes of these numb ers are irrelevant. The reduced row-echelon form

also has this feature; it only dep ends on . It is also interesting to note

of A GA

ernel has a basis consisting of orthogonal − -vectors. Finally,

that the k ( 1, 0, 1)

e see that the calculation of rank and Ker is easily done by a breadth-

w (A) (A)

h in the supp ort graph so no numerical calculation is required.

rst-searc GA

Remark. Theorem 3 and its pro of is related to the already mentioned

∗

]saying that each matrix ∈D is completely p ositive. In the pro of

result of [9 A n

this result [9] considered the graph and dened its weighted incidence

of GA

as follows. has a row for eachnodeof and a column for each matrix B B GA

1/2

in , and when ∈ , edge E b = b = a [v ,v ] E b = P A vi,[vi,vj ] vj ,[vi,vj ] i,j i j A vi,[vi,vi]

1/2

− while all other entries are zero. Then one can check that (ai,i j=6 i ai,j )

T T

In connection with the pro of of Theorem 3 we note that Ker A = BB . (B )=

T T

Ker , and that is just conditions a and b in our

Ker(BB )= (A) B x = 0 pro of.

∗

Range and linear systems. Let ∈D . Since is symmetric, wehave A n A

⊥ n

Ker where Ran { ∈ } is the range of . Thus, Ran(A)= (A) (A)= Ax : x IR A

n

consists of the vectors ∈ satisfying Ran(A) x IR

+ −

. 5

x(Cj )=x(Cj ) for j =1,...,p

consists of a single no de, the equation says that the corresp onding variable If Cj

−

zero. Cho ose, for each ≤ , an index such that ∈ if

xi is j p k(j) vk(j) Cj

let . Then, a basis of Ran consists of the vectors Cj = {vi} , k(j)=i (A)

+ − n

∈ and − , ∈ for . Let now ∈ and

ei + ek(j) , i Cj ei ek(j) i Cj j =1,...,p b IR

. This system has a solution if

consider the linear system of equations Ax = b

satises 5. Moreover, if this condition holds the solution set of and only if b

C1 Cp

span { } where is some solution of Ax = b is the ane set x0 + ( z ,...,z ) x0

Ax = b .

Positive semidenite. It is well-known that each symmetric diagonally

dominant matrix is p ositive semidenite. The fact that this is true for nonneg-

ative matrices is an immediate consequence of Prop osition 2 as wehave noted

∗

h matrix in S is p ositive semidenite or it was observed in the pro of

that eac n

of Theorem 3. The set of symmetric p ositive semidenite matrices of order

∗

vex cone PSD which contains D as a sub cone. See

n is a nonp olyhedral con n n

[5] for a discussion of many asp ects of PSD vex sets. n , related cones and con

∗

p ositive denite matrices in D maybe characterized in terms of the

The n

supp ort graph in the following way.

∗

∈D . Then the fol lowing three statements areequivalent: Let

Corollary 4 A n 5

is positive denite.

i A is nonsingular.

ii A

Each component of contains a loop or an odd cycle.

iii GA

The equivalence of i and ii follows from the fact that is p ositive

Pro of. A

Moreover, is nonsingular i rank which, by Theorem

semidenite. A (A)=n

means that , i.e., each comp onent of contains a lo op or an odd

3 p =0 GA

cycle.

For instance, consider the matrices     211 110 A =  121,B=121.

112 011

e denite b ecause is an o dd cycle a triangle while is singular

Ais p ositiv GA B

is a path. Note that these matrices are not strictly diagonally dominant,

GB P

for each and similar equations hold for . in fact, ai,i = j=6 i ai,j i B

∗

∈D b e tridiagonal, i.e., when | − | , and assume that

Let A n ai,j =0 i j > 1

. Then contains the path for

ai,i+1 = ai+1,i > 0 for i =1,...,n GA [vi,vi+1]

is connected. Thus, by Corollary 4, is nonsingular and

i =1,...,n−1,soGA A

e denite if and only if for some . Such matrices

p ositiv ai,i >ai,i+1 + ai−1,i i

are of interest in connection with cubic splines, see [10]. More generally, assume

∗

∈D is not decomp osable, i.e., there is no p ermutation matrix such that A n P

T

⊕ where and are square, nonvacuous matrices.

that P AP = A1 A2 A1 A2

alent to that is connected. Thus, Corollary

This implies in fact, it is equiv GA

es that is nonsingular and p ositive denite if and only if contains

4 giv AP GA  or an o dd cycle.

aloopai,i > j=6 i ai,j

We refer to [6], [7] and [15] for other criteria for a diagonally dominant matrix

to b e nonsingular.

aces. Recall that a nontrivial face of a convex set is the intersection F C

∗

ween and one of its supp orting hyp erplanes. Consider ∈D and let bet C A n

∗

D that contains . Then | | F (A) denote the smallest face of n A dim(F (A)) = EA

i

is the simplex cone spanned by the matrices for ∈ and and F (A) ∆ [vi,vi] EA

i,j

∈ . It follows from Theorem 3 that the maximum rank

∆ with [vi,vj] EA

is − where is the numb er of comp onents of

among the matrices in F (A) n p p

y lo op. This maximum rank is achieved

GA that are bipartite and without an

einterior of i.e., the matrices having conical

for all matrices in the relativ F (A)

tation with p ositive co ecient for each edge in . represen EA

∗

Related p olytop es. Let D with α>P0and consider the set of matrices in n

∗ ∗ n ∗

, i.e., D { ∈D } . Then D is a simplex and

trace α n(α)= A n : i=1 ai,i = α n(α)

ertices are the zero matrix and the p oints in the intersection b etween the its v P

∗ n

ys of D and the hyp erplane . Thus the nonzero vertices extreme ra n i=1 ai,i = α

∗ i i,j

D are the matrices for and for ≤ ≤ .

of n(α) α∆ i =1,...,n (α/2)∆ 1 i

ve optimization. Let real-valued convex function dened on Conca f

n,n n,n

i.e., − ≤ − for each ∈

IR , f((1 λ)A + λB) (1 λ)f(A)+λf(B) A, B IR

≤ ≤ . An example is k k where k·k is an arbitrary matrix

and 0 λ 1 f(A)= A P

norm. Another example is the linear function h i .

f(A)= C,A = i,j ci,j ai,j 6

From convexitywe know that a convex function dened on a p olytop e achieves

∗

maximum in one of the vertices, so { ∈D }equals the its max f(A):A n(α)

i i,j

um of the numb ers , for and for

maxim f(0) f(α∆ ) i =1,...,n f((α/2)∆ )

When is p ositively homogeneous, maybemoved out of the

1 ≤ i

As an example, let b e the sp ectral norm so k k is

maximization. f f(A)= A 2

value of as is symmetric. The characteristic p olynomial the largest eigen A A

i n−1 i,j n−1

is − and the characteristic p olynomial of is − . of ∆ (λ 1)λ ∆ (λ 2)λ

i i i,j i,j

es · and · .

This giv f(α∆ )=α λmax(∆ )=α f(α∆ )=(α/2) λmax(∆ )=α Therefore {k k ∈D∗ } max A 2 : A n(α) =α

i i,j

um is attained for all the matrices and .

and the maxim ∆ ∆

Matrices with nonp ositive o-diagonal elements. In the discretization

of certain partial dierential equations one is interested in symmetric diagonally

dominant matrices with nonnegative diagonal elements, but nonp ositive o-

∗

ts. Let M denote the set of such matrices of order . Using diagonal elemen n n

∗

hniques as ab ove one may show the following results. M is similar pro of tec n

i T

y the matrices a simplex cone spanned b − −

∆ as b efore and (ei ej)(ei ej)

≤ ≤ . Dene the supp ort graph nearly as b efore:

for 1 i

{ } and edges i a lo op when | | for

V = v1,...,vn [vi,vi] ai,i > j=6 i ai,j i =

when for ≤ ≤ . Again, the edges of

1,...,n and ii [vi,vj] ai,j < 0 1 i

the p ositive co ecients in the conical representation of . We corresp ond to A

∗

ve for ∈M that then ha A n

C1 Cq

span { } Ker(A)= ( χ ,...,χ )

Cj

are the comp onents of without a lo op and is the - where C1,...,Cq GA χ (0, 1)

Cj

i.e., is 1 if ∈ and 0 otherwise. Moreover, ector of

incidence v Cj χi vi Cj

− . Note that, in contrast to the case of nonnegative matrices,

rank(A)=n q

ts of are bipartite or not plays no role for the kernel

whether the comp onen GA

or the rank. But again wehave the interesting fact that the kernel and the rank dep ends only on the supp ort graph.

∗

e also see that ∈M is nonsingular if and only if each comp onentof

W A n

tains a lo op. To recognize this condition, we see that after simultaneous

GAcon

utations of rows and columns of it may b e written as the direct sum of

p erm A

say , each corresp onding to a comp onent of . smaller matrices, A1,...,Ar GA

∗

, each lies in M for some . Now, is irreducible as it corresp onds

Clearly Ai k k Ai

a comp onent of and is symmetric. Moreover, the statement that

to GA Ai P

this comp onent has a lo op just means that | | for some where

ai,i > j=6 i ai,j i

lies in that comp onent. Thus, is irreducibly diagonally dominant,

no de vi Ai

wn criterion for to b e nonsingular see [7]. a kno Ai

∗

∈M is nonsingular and for each meaning that

Note that if A n ai,i > 0 i

w, then is a Stieltjes matrix, i.e., a symmetric -matrix. A has no zero ro A M

−1

us ≥ . Th A 0

∗

values in a restricted case. Consider a matrix ∈D such that

Eigen P A n

∈{ }when 6 and for each . Thus, the supp ort graph

ai,j 0,1 i = j ai,i = j=6 i ai,j i

uniquely. We note that bychanging sign on

GA has no lo op and it determines A 7

ts of we obtain the Laplacian matrix of . Assume

all o-diagonal elemen A GA

is bipartite, say with color classes and . Thus, is singular by

that GA I J A

value of . Let denote the second

Corollary 4 and 0 is the smallest eigen A µA

value of . Then is related to connectivity prop erties of the

smallest eigen A µA

. To clarify this, note rst that supp ort graph GA X 2 (◦) µA =min{ (xi + xj ) : kxk =1, x(I)=x(J)}

[i,j]∈EA

I J

− is an eigenvector corresp onding to the eigenvalue 0  ∈ Ker ;

as z = χ χ z (A)

confer the Courant-Fischer minmax theorem, see [7]. Intro ducing the change of

ariables for ∈ and − for ∈ we see from ◦  that v yi = xi i I yi = xi i J X 2 µA =min{ (yi − yj ) : y ∈ U}

[i,j]∈EA P

n T n

consists of the vectors ∈ with k k and .

where U y IR y =1 e y = i=1 yi =1

This means that algebraic connectivity of the graph

µA is equal to the so-called

We refer to [2] for a discussion of algebraic connectivity and the Laplacian

GA .

Several prop erties of is known, but here we just mention

matrix of a graph. µA

is p ositive if and only if is connected, and ii is no greater

that i µA GA µA

yof .

than the no de connectivit GA

3 Doubly sto chastic diagonally dominant matri-

ces

A matrix is doubly sto chastic if it is nonnegative and eachrow and column sum

is 1. We let set of doubly sto chastic matrices of order . The Bn denote the n

n,n

B is a convex p olytop e in , often called the Birkho polytope. The

set n IR

on Neumann theorem states that B is the convex hull of all

classical Birkho-v n

utation matrices of order . For more information ab out this theorem and

p erm n

doubly sto chastic matrices, we refer to [2] and [7]. We are here concerned with

∗

DB of symmetric, diagonally dominant and doubly sto chastic matrices

the set n , i.e., of order n DB∗ D∗ ∩B n = n n.

∗

that DB is a convex p olytop e and that the only integral matrix in Note n

∗ ∗

DB is the identity matrix. We shall give dierent representations of DB . Let n n

∗

B be the set of symmetric matrices in B . Let denote the set of edges

n n δ(v)

t to a no de in a a graph including, p ossibly, the lo op . We need

inciden v [v, v]

the following lemma concerning the fractional p erfect matching p olytop e in

graphs with lo ops. It may b e proved using techniques explained in [12] see also

[3].

Let bea connectedgraph, possibly with loops and dene Lemma 5 G =(V,E)

E

the polytope ∈ } .

FM(G)={x∈IR : x ≥ 0 , x ( δ ( v )) = 1 for all v V 8

E

∈ is a vertex of if and only if ∈{ } for each

Then x IR FM(G) xe 0,1/2,1

dges with form node disjoint odd cycles.

e ∈ E and the e e xe =1/2

This result may b e reformulated in terms of matrices. A symmetric matrix

n,n

y be represented by a weighted graph with no des

A ∈ IR ma G =(V,E)

edges with asso ciated weight for ≤ v1,...,vn and [vi,vj] xi,j := ai,j = aj,i 1

≤ ≤ we have a lo op . We see that is symmetric i j n when i = j [vi,vi] A

E

doubly sto chastic i ∈ is nonnegative and for each and x IR x(δ(vi)) = 1

∗

≤ . Thus, B and are anely isomorphic. Let be a vertex of i n n FM(G) A

∗

B . Consider the corresp onding vertex of and cho ose an ordering

n x FM(G)

of the vertices so that i the vertices of each fractional cycle having edges

 o ccur consecutively, and ii the endno des of each edge with

with xe =1/2

ely. The no de ordering corresp onds to simultaneous line xe =1o ccur consecutiv

T

utations in and we see from Lemma 5 that the resulting matrix

p erm A Q AQ

may b e written as the direct sum of the matrices   01 (p)

[1] , ,and C 10

(p) p,p

∈ is dened by for ≤ ≤ − ,

where C =[ci,j ] IR ci,i+1 = ci−1,i =1/2 2 i p 1

otherwise. Here the rst and

c1,2 = c1,p = cp,1 = cp,p−1 =1/2and ci,j =0

the second matrix corresp onds to a lo op and an edge with ely, xe =1, resp ectiv

(p)

corresp onds to a fractional cycle of length where is o dd. This

while C p p

also shows the following result due to [8] see also [4].

∗

The set B of symmetric doubly stochastic matrices is the con- Prop osition 6 n

T

es of the form where isapermutation matrix

vex hul l of matric (1/2)(P+P ) P

der . of or n

P P

e that a matrix with satises ≥ if and Observ A ai,j =1 ai,i 6 ai,j

j ∗ j=i

it satises ≥ . It follows that DB consists of the matrices

only if ai,i 1/2 n A

satisfying the following linear system P n j=1 ai,j =1 for i =1,...,n; ≤ ≤

ai,j = aj,i for 1 i

ai,i ≥ 1/2 for i =1,...,n; ≤ ≤ ai,j ≥ 0 for 1 i

∗

descriptions of DB are contained in the following prop osition. We

Other n

denote the identity matrix of order . let In n

∗ ∗

i DB · ·B . Corollary 7 n =(1/2) In +(1/2) n

∗ T

DB is the convex hul l of the matrices · · where

ii n (1/2) In +(1/4) (P+P )

ermutation matrix of order .

P isap n 9

Pro of. Statement i is a direct consequence of 6. Next, from i we see that

∗

ertices of DB are of the form · · where is a vertex of the v n (1/2) In +(1/2) Z Z

∗

B . Thus, statement ii is a consequence of Prop osition 6. n

∗ ∗

us, the p olytop es DB and B are anely isomorphic. We note that the Th n n

∗

DB is − .

dimension of n n(n 1)/2

e get similar relations for DB D ∩B ; the diagonally dominant doubly

W n := n n

hastic matrices. So DB · ·B and DB is also equal to

sto c n =(1/2) In +(1/2) n n

vex hull of the matrices · · where is a p ermutation

the con (1/2) In +(1/2) P P

. The p olytop e DB and the Birkho p olytop e B are anely

matrix of order n n n

isomorphic.

, we p oint out that the set DB may be of interest in connection Finally n

n

[11]. If ∈ one says that is majorized by , with ma jorization see x, yP IR P y x

k k

≺ , provided that ≤ for − by denotedP y Px j=1 y[j] j=1 x[j] k =1,...,n 1

n n

. Here denotes the th largest numb er among the

and j=1 xj = j=1 yj x[j] j

ts of . Awell-known theorem of Hardy-Littlewo o d and Pólya see

comp onen x

] says that ≺ if and only if for some ∈B . Consider now the

[11 y x Bx = y B n

stronger prop erty that ∈DB 7

Ax = y for some A n

is not just doubly sto chastic, but also diagonally dominant. From the

so A

DB given ab ovewe see that 7 holds if and only if

description of n y =(1/2)x+

≺ . The geometrical interpretation is that is the midp oint

(1/2)z for some z x y

tbetween and a p oint in the convex hull of all p ermutations

of the line segmen x z

. Or, equivalently, is a convex combination of p oints of the form

of x y (1/2)x +

is a p ermutation matrix. A similar characterization maybe (1/2)Px where P

∗

en when 7 holds for a matrix in DB using Prop osition 6 and Corollary

giv n

7.

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