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A Note on Nonnegative Diagonally Dominant Matrices Geir Dahl

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A Note on Nonnegative Diagonally Dominant Matrices Geir Dahl 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:[email protected] 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<j n ¯ 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<j n Xn X i i,j (∗) A = λi∆ + λi,j ∆ . i=1 i<j 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<j n 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<i λj,i + − . From this we conclude that i<j λi,j = λi + j=6 i ai,j so λi = ai,i j=6 i ai,j ∗ 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<j ∗ 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<j n 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 .
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