A property concerning the Hadamard powers of inverse M-matrices

Linear Algebra and its Applications 381 (2004 53 60 www.elsevier.com/locate/laa A property concerning the Hadamard powers of inverse M-matrices Shencan Chen Department of Mathematics, Fuzhou University, Fuzhou, Fujian 350002, PR China Received 23 April 2003; accepted 22 September 2003 Submitted by R.A. Brualdi Abstract The main result of this paper is the following: if A = (a ij is an inverse M-matrix, A ( = (a ij denotes the th Hadamard power of A,thenA( is an inverse M-matrix for any positive integer >1. In the case = 2, this settles a conjecture of Neumann [Linear Algebra Appl. 285 (1998 278] affirmatively. 2004 Published by Elsevier Inc. AMS classification: 15A09; 15A42 Keywords: M-matrix; Inverse M-matrix; Hadamard product 1. Introduction All matrices considered in this paper are real. A matrix is called nonnegative if every entry is nonnegative. For two m n matrices A = (a ij and B = (b ij, A B means that A B is nonnegative, the Hadamard product of A B is defined and denoted by A B = (a ij b ij.forα 0, we write A (α = (aij α for the αth Hadamard power of the matrix A. For a positive integer n, letn ={1, 2,...,n} throughout. To avoid triviality, we always assume that n>1. Given an n n matrix A, for nonempty index set α, β N,wedenotebyA(α, β that submatrix of A lying in the rows indicated by α and the columns indicated by Supported by Foundation to the Educational Committee of Fujian, China (grant no. JB02084 and the Science and Technical Development Foundation of Fuzhou University (2003-XQ-22. E-mail address: shencan@public.fz.fj.cn (S. Chen. 0024-3795/$ - see front matter 2004 Published by Elsevier Inc. doi:10.1016/j.laa.2003.09.021

54 S. Chen / Linear Algebra and its Applications 381 (2004 53 60 β; the principal submatrix A(α, α is abbreviated to A(α, and for brevity A(α, {i} and A({i},αare denoted by A(α, i and A(i, α respectively. Let A be an n n matrix. A is called a Z-matrix if all of whose off-diagonal entries are nonpositive; A is called an M-matrix if A is a Z-matrix and A 1 0; A is called an inverse M-matrix if A 1 is an M-matrix. It is well nown that an inverse M-matrix is nonnegative. It follows that a nonsingular nonnegative matrix is an inverse M-matrix if and only if its inverse is a Z-matrix. We shall mae frequent use of this observation. In [1, p. 278], Neumann conjectured a property concerning the Hadamard product of inverse M-matrices as follows: If A is an n n matrix which is an inverse of an M-matrix, then A A is an inverse of an M-matrix. It is nown, via an example due to Johnson, Marham, and Neumann (see [2], that the more general conjecture, that if A and B are inverse M-matrices of the same size, then A B is an inverse M-matrix, is false. Neumann [1] has showed that the conjecture is true for several well-nown classes of inverse M-matrices. In this paper, we obtain a special property concerning the Hadamard powers of inverse M-matrices: If A is an n n inverse M-matrix, then A ( is an inverse M-matrix for any positive integer >1. When = 2, this shows that the conjecture of Neumann is valid. 2. Main results Basic for our purpose is the following simple facts, we will use later. Lemma 2.1. Let A be an n n inverse M-matrix. Then: (a All principal minors of A are positive. (b Each principal submatrix of A is an inverse M-matrix. (c For any permutation matrix P of order n, P AP T is an inverse M-matrix. (d For any nonempty proper subset α of N, A(α 1 A(α, α c 0, A(α c,α A(α 1 0, where α c = N\α. Proof. The conclusions (a and (b can be found in [3]; the assertion (d can be found in [4, p. 254], and the result (c is obvious. Lemma 2.2 (a If A is an n n inverse M-matrix, then adj A(the adjoint of A is a Z-matrix. (b Let A = (a ij be an n n nonnegative matrix,n 3. Then the following conditions are equivalent:

S. Chen / Linear Algebra and its Applications 381 (2004 53 60 55 (i Each principal minor of A of order 2 is nonnegative, and adj B is a Z-matrix for any principal submatrix B of A of order 3. (ii a i a j a a ij for all i, j, in N. (1 Proof. (a If A is an inverse M-matrix, then A>0, by Lemma 2.1(a. Since A 1 = 1 Aadj A is an M-matrix, thus adj A is a Z-matrix. (b (ii (i: This is nearly trivial; (i (ii: Suppose the condition (i holds i, j, N, we distinguish the following two cases: Case 1: (i j(j ( i = 0. Since all principal minors of A of order 2 are nonnegative, it is easy to chec that (1 holds in this case. Case 2: (i j(j ( i /= 0. We set a ii a ij a i B = a ji a jj a j. a i a j a By our assumption, there exists a permutation matrix P of order 3 such that adj(p T BP is a Z-matrix. Since adj(p T BP = (adj P(adj B(adj P T = ( PP 1 (adj B(P T 1 P T = P T (adj BP,adjB = P [adj(p T BP]P T is also a Z-matrix. This yields that the (1, 2th entry of adj B is nonpositive, namely, (a ij a a i a j 0, therefore, a i a j a a ij. The condition (ii holds. This completes the proof. Given an n n nonnegative matrix A = (a ij, n 3. It is nown that the inequality (1 is necessary for A to be an inverse M-matrix [5, Theorem 1]. But if the inequality (1 holds, we cannot imply that A is an inverse M-matrix, as can be illustrated by the following matrix: A = (a ij with a ij = 1foralli, j in N. Thus Lemma 2.2(b represents an improvement of Willoughby s observation. It will be used to prove our Theorem 2.4. Lemma 2.3. Let A be an inverse M-matrix of order n, whose columns are denoted by α 1,α 2,...,α n. Then for any x = (x 1,x 2,...,x n T, the functions f(x= (α 1,α 2,...,α n 1,x and g(x = (x, α 2,...,α n 1,α n have the following properties: (a If x = (x 1,x 2,...,x n T y = (y 1,y 2,...,y n T, and x n = y n, then f(x f(y. (b If x = (x 1,x 2,...,x n T y = (y 1,y 2,...,y n T, and x 1 = y 1, then g(x g(y. Proof. Using Lemma 2.2(a, adj A (b ij is a Z-matrix. Observe that b ij 0 ( i, j N,i /= j, we have

56 S. Chen / Linear Algebra and its Applications 381 (2004 53 60 (a If x n = y n, x = (x 1,x 2,...,x n T y = (y 1,y 2,...,y n T,then n 1 n 1 f(x= x i b ni + x n b nn y i b ni + y n b nn = f(y. i=1 (b If x 1 = y 1, x = (x 1,x 2,...,x n T y = (y 1,y 2,...,y n T,then n n g(x = x i b 1i + x 1 b 11 y i b 1i + y 1 b 11 = g(y. i=2 i=1 i=2 Now we provide a new sufficient and necessary condition for a square nonnegative matrix to be an inverse M-matrix as follows. Theorem 2.4. Let A = (a ij be an n n nonnegative matrix, n 3. Then the following statements are equivalent: (a A is an inverse M-matrix. (b adj A is a Z-matrix, and each proper principal submatrix of A is an inverse M-matrix. Proof. In view of Lemma 2.1(b and Lemma 2.2(a, it suffices to show that (b implies (a. Indeed, if (b holds, then using Lemma 2.2(b, we now that a i a j a a ij ( i, j, N. Put δ = N\{1,n}, andleta be partitioned as a 11 A(1,δ a 1n A = A(δ, 1 A(δ A(δ, n. a n1 A(n, δ a nn Let adj A (b ij. Since adj A is a Z-matrix, we have b 1n = ( 1 n+1 A(1,δ a1n a1n A(1,δ = 0, A(δ A(δ, n A(δ, n A(δ b n1 = ( 1 1+n A(δ, 1 A(δ A(δ A(δ, 1 = 0. a n1 A(n, δ Hence a1n A(1,δ A(δ A(δ, 1 0, 0. A(δ, n A(δ Since a i1 a 1n a 11 a in, a in a n1 a nn a i1 ( i δ, we obtain a 1n A(δ, 1 a 11 A(δ, n, a n1 A(δ, n a nn A(δ, 1. Observe that each principal submatrix of A of order n 1isaninverseM-matrix, according to Lemma 2.3, we deduce that

S. Chen / Linear Algebra and its Applications 381 (2004 53 60 57 a11 A(1,δ a11 a a 1n = 1n A(1,δ A(δ, 1 A(δ a 1n A(δ, 1 A(δ a11 a 1n A(1,δ a 11 A(δ, n A(δ a1n A(1,δ = a 11 0, A(δ, n A(δ A(δ A(δ, n A(δ an1 A(δ, n a n1 = A(n, δ a nn a nn A(δ ann A(δ, 1 A(n, δ a nn a n1 = a nn A(δ A(δ, 1 0. By above obtained inequalities, we infer that A(1,δ a1n A(δ, 1 A(δ A(δ A(δ, n a n1 A(n, δ = ( 1 n 2 a1n A(1,δ ( 1 n 2 A(δ A(δ, 1 A(δ, n A(δ = 1 a1n A(1,δ A(δ A(δ, 1 a 11 a a 11 a nn A(δ, n A(δ nn a 1na n1 a11 A(1,δ A(δ A(δ, n. a 11 a nn A(δ, 1 A(δ A(n, δ a nn Applying Eq. (1.5 of [4], it follows that a11 A(1,δ A(δ A(δ, n A A(δ = A(δ, 1 A(δ A(n, δ a nn A(1,δ a1n A(δ, 1 A(δ A(δ A(δ, n a n1 A(n, δ ( 1 a 1na n1 a11 A(1,δ a 11 a nn A(δ, 1 A(δ A(δ A(δ, n A(n, δ a nn > 0 (by our assumption and Lemma 2.1(a Consequently, A>0. Since adj A is a Z-matrix, and A 1 = 1 is a Z-matrix. The proof is completed. A adj A,thenA 1

58 S. Chen / Linear Algebra and its Applications 381 (2004 53 60 Given a nonnegative matrix A = (a ij of order 3. By examining the nown proof of Theorem 1 of [5] carefully, we now that A is an inverse M-matrix if and only if the following conditions are satisfied for (i,j,any permutation of (1,2,3: (i a ii > 0; (ii a i a j a a ij ; (iii a ii a >a i a i. When n = 3, it is not difficult to see that above-mentioned conditions are equivalent to the condition (b of Theorem 2.4. Thus Theorem 2.4 denotes a generalization of Willoughby s result. To prove our main result Theorem 2.6, we must apply this criterion. Lemma 2.5. Let A = (a ij be an n n inverse M-matrix, whose columns are denoted by α 1,α 2,...,α n. For any positive integer >1, if A ( is an inverse M- matrix, then for all,i 2,...,i in N, (,i 2,...,i = ( α ( 1,α( 2,...,α( n 1,α α i2 α i 0. Proof. When n = 2, since a 11 a 2i a 21 a 1i 0 (i = 1, 2, wehave a (,i 2,...,i = 11 a 1i1 a 1i2 a 1i a 2i1 a 2i2 a 2i a 21 = (a 11 a 2i1 (a 11 a 2i2 (a 11 a 2i (a 21 a 1i1 (a 21 a 1i2 (a 21 a 1i 0 (,i 2,...,i N. Below we assume that n 3,,i 2,...,i N, wedefine B ( α ( 1,α( 2,...,α( n 1,α α i2 α i (bij. We consider two possibilities as follows: Case 1: a ni1 a ni2 a ni = 0 (1,i 2,...,i n Let S n be the set of all permutations of N. Forallσ S n, we claim that b 1σ(1 b 2σ(2 b nσ (n = 0. (2 In fact, if σ (n = n, thenb nσ (n = a ni1 a ni2 a ni = 0, thus (2 holds. If σ (n /= n, wetaes 1 = n, andletr = min{t : σ t (n = n, t N}, thenr>1, and form an r-cycle: σ(s 1 = s 2,...,σ(s r 1 = s r, σ(s r = s 1. Applying the inequality (1, repeatedly, we deduce that b s1 s 2 b sr 1 s r b sr s 1 = as 1 s 2 as r 1 s r (a sr a sr i 2 a sr i = (a s1 s 2 a sr 1 s r a sr (a s1 s 2 a sr 1 s r a sr i 2 (a s1 s 2 a sr 1 s r a sr i (a s2 s 2 a sr s r (a s1 a s1 i 2 a s1 i = 0.

S. Chen / Linear Algebra and its Applications 381 (2004 53 60 59 By our assumption, this implies that b s1 s 2 b sr 1 s r b sr s 1 = 0. Because σ is a product of some disjoint cycles, this means that (2 holds. Moreover, this gives rise to (,i 2,...,i = B = σ s n sgn(σ b 1σ(1 b 2σ(2 b nσ (n = 0. Case 2: a ni1 a ni2 a ni /= 0(1,i 2,...,i n We put c 1 = a n a ni2 a ni ani,c 2 = a n a ni2 a ni 1 ani,...,c = a n a ni2 a ni 2 ani. Applying the inequality of the arithmetic and geometric means, we have 1( c1 ali 1 + c 2 ali 2 + +c ali ali1 a li2 a li ( l N with equality for l = n. Soα i1 α i2 α i c 1 α ( + c 2 α ( i 2 + + c α ( i. α ( n Observe that A ( is an inverse M-matrix, by Lemma 2.1(a, (α ( 1,α( >0. According to Lemma 2.3, we infer that (,i 2,...,i = t=1 ( α ( 1,...,α( n 1, c 1 We now turn to prove the main result. α( + c 2 α( i 2 c t ( α ( 1,...,α( n 1,α( i t 0. 2,..., + + c α( i Theorem 2.6. If A is an inverse M-matrix of order n, then A ( is an inverse M- matrix for any positive integer >1. Proof. We proceed by induction on n. It is not difficult to verify that our assertion is true for n = 2. Now we assume that n 3, and the assertion is true for all inverse M-matrices of size at most n 1. Let b ij denote the (i, jth entry of adj A (, then it is easy to see that ( [A(σ ] ( [A(σ, j] ( b ij = [A(i, σ ] (, a ij where i, j N, i/= j, σ = N\{i, j}. Define A(σ A(σ, i A(σ, j B =, A(i, σ a ii a ij

60 S. Chen / Linear Algebra and its Applications 381 (2004 53 60 and let β 1,β 2,...,β n 1,β n be the columns of B. By Lemma 2.1(c, the matrix A(σ A(σ, i A(σ, j H = A(i, σ a ii a ij A(j, σ a ji a jj is an inverse M-matrix. In view of Lemma 2.1(d, we have (β 1,β 2,...,β n 1 1 β n = (x 1,x 2,..., x n 1 T 0, thus, β n = n 1 i=1 x iβ i, and hence β ( n 1 n 1 n 1 x i1 β i1 x i2 β i2 x i β i n = = =1 1,i 2,...,i n 1 i 2 =1 i =1 x i1 x i2 x i (β i1 β i2 β i. (3 By the induction hypothesis, we now that each proper principal submatrix of H ( is an inverse M-matrix. In particular, the matrix (β ( 1,β( 2,...,β( n 2,β( n 1 is an inverse M-matrix of order n 1. Using Eq. (3, we deduce that b ij = ( β ( 1,β( 2,...,β( n 2,β( n = x i1 x i2 x i ( β ( 1,β( 2,...,β( n 2,β β i2 β i 1,i 2,...,i n 1 0 (by Lemma 2.5 whence adj A ( is a Z-matrix. Finally, Theorem 2.4 ensures that A ( is an inverse M-matrix, and the result follows. Acnowledgement I than the referee for many valuable suggestions which improved our presentation. References [1] M. Neumann, A conjecture concerning the Hadamard product of inverses of M-matrices, Linear Algebra Appl. 285 (1998 277 290. [2] C.R. Johnson, Closure properties of certain positivity classes of matrices under various algebraic operations, Linear Algebra Appl. 97 (1987 243 247. [3] C.R. Johnson, Inverse M-matrices, Linear Algebra Appl. 47 (1982 195 216. [4] C.R. Johnson, R.L. Smith, Almost principal minors of inverse M-matrices, Linear Algebra Appl. 337 (2001 253 265. [5] R.A. Willoughby, The inverse M-matrix problem, Linear Algebra Appl. 18 (1977 75 94. [6] B.Y. Wang, X.P. Zhang, F.Z. Zhang, On the Hadamard products of inverse M-matrices, Linear Algebra Appl. 305 (2000 23 31.