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Ray pattern matrices requiring Perron–Frobenius properties
Linear Algebra and its Applications 571 (2019) 103–109
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
Ray pattern matrices requiring Perron–Frobenius properties ✩ Yue Liu ∗ , Lidan Tang College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 350116, China
a r t i c l e
i n f o
Article history: Received 16 October 2018 Accepted 23 February 2019 Available online 28 February 2019 Submitted by R. Brualdi MSC: 15B35 15B48 Keywords: Weak/strong Perron–Frobenius property Ray pattern Diagonally similar Positive cycle
a b s t r a c t A ray pattern matrix is a matrix whose nonzero entries are all unimodular. An n × n ray pattern matrix A is said to require the weak Perron–Frobenius property if for any n ×n entrywise positive matrix K, the spectral radius ρ(K ◦ A) is one of the eigenvalues of K ◦ A; to require the strong Perron–Frobenius property if ρ(K ◦ A) is an algebraically simple eigenvalue of K ◦A and the left and right eigenvectors associated to ρ(K ◦A) are strictly nonzero. In this paper, necessary and sufficient conditions for ray pattern matrices requiring the weak/strong Perron–Frobenius property are given. © 2019 Elsevier Inc. All rights reserved.
1. Introduction A complex matrix is called a ray pattern matrix if all of its nonzero entries are unimodular. The ray pattern of a complex matrix A, denoted by ray(A), is the ray pattern matrix obtained from A by replacing each nonzero entry ajk by ajk /|ajk |. A is a non✩
This research is supported by National Natural Science Foundation of China 11571075 and 11601084.
* Corresponding author. E-mail addresses: [email protected] (Y. Liu), [email protected] (L. Tang). https://doi.org/10.1016/j.laa.2019.02.020 0024-3795/© 2019 Elsevier Inc. All rights reserved.
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negative real matrix is equivalent to saying that ray(A) is a 0-1 matrix. The concept of ray pattern matrices is a generalization of that of sign pattern matrices (see [1]) from the real field to complex field. In the study of nonnegative matrix theory, the Perron–Frobenius Theorem is one of the most important theorems, and it has wide applications that can be found in many monographs on matrix theory. Specially, in [2], we have: Theorem 1.1. ([2, Theorem 8.4.4]) Let A ∈ Mn be irreducible and nonnegative, and suppose that n ≥ 2. Then (a) ρ(A) > 0. (b) ρ(A) is an algebraically simple eigenvalue of A. (c) there is a unique real vector x = [xi ] such that Ax = ρ(A)x and x1 + · · · + xn = 1; this vector is strictly positive. (d) there is a unique real vector y = [yi ] such that y T A = ρ(A)y T and x1 y1 +· · ·+xn yn = 1; this vector is strictly positive. In the above theorem, Mn is the set of complex square matrices with order n, and ρ(A) is the spectral radius of A. In fact, if there is no restriction of the irreducibility, the Perron–Frobenius Theorem tells us that the spectral radius of a nonnegative matrix is always one of its eigenvalues. We say a square matrix A has the weak Perron–Frobenius property if ρ(A) is one of its eigenvalues. Furthermore, A is said to have the strong Perron–Frobenius property if ρ(A) is an algebraically simple eigenvalue of A and the left and right eigenvectors associated to ρ(A) are strictly nonzero. Matrices having the weak/strong Perron–Frobenius property are not necessarily be nonnegative, for example, see [3–6] and other related articles. Most of these studies are concentrated in the real field. In this paper, we discuss the problem over the complex field. Just as one natural way to represent a nonzero complex number z is z = reiθ , where eiθ is unimodular and r is positive, a complex matrix A can be represented as A = K ◦ ray(A), where K is an entry-wise positive matrix, and ◦ is the Hadamard product. We are interested in the problem of what properties of the complex matrix A can be determined just by ray(A). An n ×n ray pattern matrix A is said to require the weak (or strong) Perron–Frobenius property if for every n ×n entry-wise positive matrix K, K ◦A has the weak (or strong, respectively) Perron–Frobenius property. Then part of the Perron–Frobenius Theorem can be restated as: 0-1 square ray pattern matrices require the weak Perron–Frobenius prop-
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erty, and irreducible 0-1 square ray pattern matrices require the strong Perron–Frobenius property. Let Q(A) = {B| ray(B) = ray(A)}. Q(A) is called the ray pattern class of A. Then A requires the weak (strong) Perron–Frobenius property means for every matrix B ∈ Q(A), B has the weak (strong, respectively) Perron–Frobenius property. A natural problem arises: Question: Besides the 0-1 ray pattern matrices, are there any other ray pattern matrices that require the weak/strong Perron–Frobenius property? How does one characterize these ray pattern matrices? In [7], it is shown that if all of the cycles in the associated digraph of a square complex matrix A are negative, then A is diagonally similar to an SNS matrix. In this paper, we use a similar method to show that a ray pattern matrix requiring the weak Perron–Frobenius property if and only if it is diagonally similar to a 0-1 matrix; and that it requires the strong Perron–Frobenius property if and only if it is diagonally similar to an irreducible 0-1 matrix. 2. Main result Let A = (ajk )n×n be a complex square matrix with order n. Then A will sometimes be identified with its associated arc-weighted digraph WA = (D, w), where D is the digraph with the vertex set [n] = {1, 2, . . . , n}, for every j, k ∈ [n], (j, k) is an arc of D if and only if ajk = 0, and w((j, k)) = ajk . Note that there is no multi-arc in D, but it’s possible there are some loops in D. Loops are treated as directed cycles with length 1. (In this paper, all the cycles in digraph are understood to be directed cycles.) The cycles in WA are also called the cycles of A. Let C : j1 → j2 → . . . → js → j1 be a cycle of A, then C is called a positive cycle if the product aj1 j2 aj2 j3 · · · ajs j1 is positive; this product is also called the weight of the cycle C, and denoted by w(C). Lemma 2.1. Let A ∈ Mn . If the arc-weighted digraph of A is a cycle C with length s, and w(C) = w, then the spectrum of A consists of n − s copies of 0 and all s roots of the equation λs − w = 0. Proof. The conclusion follows from the facts that the characteristic polynomial of A is |λI − A| = λn − λn−s w = λn−s (λs − w). 2 To prove the necessity of our result, we need the following continuity theorem of the matrix spectrum with respect to its entries. Theorem 2.1. ([2, Theorem 2.4.9.2]) Let an infinite sequence A1 , A2 , . . . ∈ Mn be given, and suppose that limk→∞ Ak = A (entrywise convergence). Let λ(A) =
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[λ1 (A) . . . λn (A)]T and λ(Ak ) = [λ1 (Ak ) . . . λn (Ak )]T be given presentations of the eigenvalues of A and Ak , respectively, for k = 1, 2, . . .. Let Sn = {π : π is a permutation of {1, 2, . . . , n}} . Then for each given > 0 there exists a positive integer N = N () such that min
max
π∈Sn i=1,2,...,n
|λπ(i) (Ak ) − λi (A)| ≤ for all k ≥ N
Two square matrices A and B are said to be diagonally similar if there exists a diagonal invertible matrix Q = diag(q1 , q2 , . . . , qn ) such that Q−1 AQ = B. It is easy to see diagonal similarity is an equivalence relationship. Furthermore, from the view of graph theory, if WA and WB are the arc-weighted digraphs corresponding to A and B, respectively, then B = Q−1 AQ is equivalent to the statement that WB can be obtained from WA by multiplying the weights of all the arcs going out of vertex j by qj and multiplying the weights of all the arcs going into j by qj−1 for j = 1, . . . , n. We call this kind of graph operation an arc weight redistribution. The following lemma gives a necessary and sufficient condition for the diagonal similarity, it is a generalization of Theorem 4.1 in [7]. Also, the proof idea is similar. Lemma 2.2. Let A = (ajk )n×n , B = (bjk )n×n be complex irreducible square matrices, WA = (DA , wA ) and WB = (DB , wB ) be the arc-weighted digraphs corresponding to A and B, respectively. Then A is diagonally similar to B if and only if DA = DB , and for every cycle C in DA (and DB ), wA (C) = wB (C). Proof. Suppose that B = Q−1 AQ, where Q = diag(q1 , q2 , . . . , qn ). It is easy to see bjk = qj−1 ajk qk . Then ajk = 0 if and only if bjk = 0, thus DA = DB . Let C : j1 → j2 → · · · → js → j1 be a cycle in D. Then wB (C) = bj1 j2 bj2 j3 · · · bjs j1 = qj−1 aj1 j2 qj2 qj−1 aj2 j3 qj3 · · · qj−1 a js j1 q j1 1 2 s = wA (C). Now we use a method of graph theory to prove the sufficiency part. Note that A, B are irreducible, then DA (= DB ) are strongly connected. By the recursive constructions of strongly connected digraphs (see [1, P108]), we can assume that DA = D + P , where D is a strongly connected sub-digraph of DA , and P : x → k1 → k2 → · · · → kt → y is a directed path (or cycle) connecting two vertices x and y in D (if x = y, then P is a cycle). Without loss of generality, we may assume the vertex set of D is [m], where m is the order of D . Let WA (WB ) be the sub-digraph of WA (WB , respectively) whose underlying digraph is D .
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By induction on the number of arcs in DA , WB can be obtained by a proper arc weight redistribution from WA , now apply the same arc weight redistribution operation on the vertex set [m] of WA , and denote the result arc-weighted digraph by W = (DA , w). Then the only possible difference between WB and W are the weights of the arcs on the path P . By successively applying proper arc weight redistribution operations on the vertices k1 , k2 , . . . , kt , we can further assume that w((x, k1 )) = wB ((x, k1 )), w((k1 , k2 )) = wB ((k1 , k2 )), . . . , w((kt−1 , kt )) = wB ((kt−1 , kt )). For the last arc (kt , y), w(kt , y) has to be the same as wB (kt , y) since (kt , y) needs to belong to some cycle C in DA and w(C) = wB (C) for every cycle in DA . Then W = WB , which means WB can be obtained from WA by a suitable arc weight redistribution operation, yielding that A and B are diagonally similar. 2 Theorem 2.2. Let A be a square ray pattern matrix with order n. Then (a) A requires the weak Perron–Frobenius property if and only if the weight of every cycle in A is positive. (b) A requires the strong Perron–Frobenius property if and only if WA is strongly connected and the weight of every cycle in A is positive. Proof. (1) We first prove the necessity part. Suppose to the contrary that there exists a cycle C : j1 → j2 → · · · → js → j1 such that wA (C) = eiθ , where θ ∈ (0, 2π). Write K = K() = (kjs )n×n , where kjs = 1 if (j, s) ∈ C and kjs = otherwise. Write B = B() = K ◦ A. By Theorem 2.1 and Lemma 2.1, ρ(B) → 1 as → 0+ , but none of B’s eigenvalues will converge to 1. Thus for > 0 small enough, ρ(B) will not be an eigenvalue of B, contradicting the fact that A requires the Perron–Frobenius property. For the sufficiency part, after applying suitable simultaneous row and column permutations, without loss of generality we assume that A is a upper triangle block matrix whose diagonal blocks A1 , A2 , . . . Ar are all irreducible. For any entrywise positive matrix K, write B = K ◦ A, and B1 , B2 , . . . , Br to be the diagonal irreducible blocks of B. For each j = 1, 2, . . . , r, let Cj be the matrix obtained from Bj by replacing each of its nonzero entries with the corresponding modulus. Then the underlying digraph of Bj and Cj are the same. Since all the cycles in A are positive, then all the cycles in B1 , B2 , . . . , Br are also positive. Thus by Lemma 2.2, B1 , B2 , . . . , Br are diagonally similar to C1 , C2 , . . . , Cr , respectively. Since diagonally similar matrices share the same spectrum, then the ρ(Bj ) is an eigenvalue of Bj for every j = 1, 2, . . . , r by the Perron–Frobenius Theorem. Note that ρ(B) = maxj ρ(Bj ), and spectrum of B is the union of the spectra of B1 , B2 , . . . , Br , then ρ(B) is still an eigenvalue of B, thus A requires the weak Perron–Frobenius property. (2) It is well known that the irreducibility of A is equivalent to the strong connectivity of WA , so together with (1) and the Perron–Frobenius Theorem, the sufficiency part holds.
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For the necessity part, we only need to prove that A is irreducible. Suppose to the contrary, without loss of generality, assume that A is a upper triangle block matrix whose diagonal blocks A1 , A2 , . . . Ar are all irreducible. Let B = K ◦ A, B1 , B2 , . . . , Br be the diagonal blocks of B, respectively. By properly choosing K, we can assume that ρ(B) = ρ(B1 ) = ρ(B2 ), which means ρ(B) is not algebraically simple, contradicting to that A requires strong Perron–Frobenius property. 2 The Perron–Frobenius Theorem can be generalized from real field to complex field as below: Theorem 2.3. Let A ∈ Mn (not necessarily be a ray pattern matrix), n ≥ 2. If A is irreducible and all the cycles in A are positive, then (a) (b) (c) (d) (e)
ρ(A) > 0. ρ(A) is an algebraically simple eigenvalue of A. there is an entrywise nonzero vector x = [xj ] such that Ax = ρ(A)x. there is an entrywise nonzero vector y = [yj ] such that y T A = ρ(A)y T . xj yj are all positive for j = 1, 2, . . . , n.
Proof. Let B be the matrix obtained from A by replacing each of nonzero entries with the corresponding modulus. Then B is a irreducible nonnegative matrix. By Lemma 2.2, there exists an invertible matrix Q = diag(q1 , q2 , . . . , qn ) such that B = Q−1 AQ. By the Perron–Frobenius Theorem, corresponding to its eigenvalue ρ(B), B has an (B) entrywise positive right eigenvector x(B) = [xj ], and an entrywise positive left eigen(B)
vector y (B) = [yj
]. Choose x = Qx(B) , y = Q−1 y (B) , then x and y are the right and left
eigenvectors of A corresponding to ρ(A), respectively; and xj = qj xj , yj = qj−1 yj so xj yj is positive for every j = 1, 2, . . . , n, completing the proof. 2 (B)
(B)
,
Next we restrict our attention to ray pattern matrices. An n × n complex matrix S is defined to be a signature matrix if S is a diagonal matrix all of whose diagonal entries are unimodular. Note that S −1 = S¯ is still a signature matrix. An n × n complex matrix M is defined to be signature nonnegative (respectively, signature positive) if there exists an n × n signature matrix S such that SM S −1 is entrywise nonnegative (respectively, entrywise positive). We have the following two results. Theorem 2.4. (a) An n × n irreducible ray pattern matrix A is signature nonnegative if and only if all cycles in the arc-weighted directed graph WA are positive. (b) An n × n entrywise nonzero square ray pattern matrix A is signature positive if and only if the arc-weighted directed graph WA has all cycles be positive.
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Proof. The implication parts holds trivially for both of these results. We only need to prove the inverse implication parts. (1) Let B be the matrix obtained from A by replacing each of nonzero entries with 1. Since all the nonzero entries of A are unimodular, then for every cycle C in DA (= DB ), wA (C) = wB (C) = 1. By Lemma 2.2, there exists an invertible matrix S = diag(s1 , s2 , . . . , sn ) such that A = SBS −1 . Note that in the proof of Lemma 2.2, the arc weight redistribution operations do not change the modulus of the arc weights, thus s1 , s2 , . . . , sn are all unimodular, yielding that S is a signature matrix. The proof of (2) is similar. 2 Acknowledgements The authors would like to thank the referee for his/her careful reading, valuable suggestions and encouragement! Also, Theorem 2.4 is proposed by the referee. References [1] R.A. Brualdi, B.L. Shader, Matrices of Sign-Solvable Linear Systems, Vol. 116, Cambridge University Press, 1995. [2] R.A. Horn, C.R. Johnson, Matrix Analysis, second edition, Cambridge University Press, 2012. [3] C.R. Johnson, P. Tarazaga, On matrices with Perron–Frobenius properties and some negative entries, Positivity 8 (4) (2004) 327–338. [4] D. Noutsos, On Perron–Frobenius property of matrices having some negative entries, Linear Algebra Appl. 412 (2) (2006) 132–153. [5] P. Tarazaga, M. Raydan, A. Hurman, Perron–Frobenius theorem for matrices with some negative entries, Linear Algebra Appl. 328 (1) (2001) 57–68. [6] A. Elhashash, D.B. Szyld, On general matrices having the Perron–Frobenius property, Electron. J. Linear Algebra 17 (1) (2008) 389–413. [7] J.-Y. Shao, H.-Y. Shan, L.-H. You, Ray solvable linear systems and ray S 2 N S matrices, Linear Algebra Appl. 395 (2005) 229–246.
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
Ray pattern matrices requiring Perron–Frobenius properties ✩ Yue Liu ∗ , Lidan Tang College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 350116, China
a r t i c l e
i n f o
Article history: Received 16 October 2018 Accepted 23 February 2019 Available online 28 February 2019 Submitted by R. Brualdi MSC: 15B35 15B48 Keywords: Weak/strong Perron–Frobenius property Ray pattern Diagonally similar Positive cycle
a b s t r a c t A ray pattern matrix is a matrix whose nonzero entries are all unimodular. An n × n ray pattern matrix A is said to require the weak Perron–Frobenius property if for any n ×n entrywise positive matrix K, the spectral radius ρ(K ◦ A) is one of the eigenvalues of K ◦ A; to require the strong Perron–Frobenius property if ρ(K ◦ A) is an algebraically simple eigenvalue of K ◦A and the left and right eigenvectors associated to ρ(K ◦A) are strictly nonzero. In this paper, necessary and sufficient conditions for ray pattern matrices requiring the weak/strong Perron–Frobenius property are given. © 2019 Elsevier Inc. All rights reserved.
1. Introduction A complex matrix is called a ray pattern matrix if all of its nonzero entries are unimodular. The ray pattern of a complex matrix A, denoted by ray(A), is the ray pattern matrix obtained from A by replacing each nonzero entry ajk by ajk /|ajk |. A is a non✩
This research is supported by National Natural Science Foundation of China 11571075 and 11601084.
* Corresponding author. E-mail addresses: [email protected] (Y. Liu), [email protected] (L. Tang). https://doi.org/10.1016/j.laa.2019.02.020 0024-3795/© 2019 Elsevier Inc. All rights reserved.
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negative real matrix is equivalent to saying that ray(A) is a 0-1 matrix. The concept of ray pattern matrices is a generalization of that of sign pattern matrices (see [1]) from the real field to complex field. In the study of nonnegative matrix theory, the Perron–Frobenius Theorem is one of the most important theorems, and it has wide applications that can be found in many monographs on matrix theory. Specially, in [2], we have: Theorem 1.1. ([2, Theorem 8.4.4]) Let A ∈ Mn be irreducible and nonnegative, and suppose that n ≥ 2. Then (a) ρ(A) > 0. (b) ρ(A) is an algebraically simple eigenvalue of A. (c) there is a unique real vector x = [xi ] such that Ax = ρ(A)x and x1 + · · · + xn = 1; this vector is strictly positive. (d) there is a unique real vector y = [yi ] such that y T A = ρ(A)y T and x1 y1 +· · ·+xn yn = 1; this vector is strictly positive. In the above theorem, Mn is the set of complex square matrices with order n, and ρ(A) is the spectral radius of A. In fact, if there is no restriction of the irreducibility, the Perron–Frobenius Theorem tells us that the spectral radius of a nonnegative matrix is always one of its eigenvalues. We say a square matrix A has the weak Perron–Frobenius property if ρ(A) is one of its eigenvalues. Furthermore, A is said to have the strong Perron–Frobenius property if ρ(A) is an algebraically simple eigenvalue of A and the left and right eigenvectors associated to ρ(A) are strictly nonzero. Matrices having the weak/strong Perron–Frobenius property are not necessarily be nonnegative, for example, see [3–6] and other related articles. Most of these studies are concentrated in the real field. In this paper, we discuss the problem over the complex field. Just as one natural way to represent a nonzero complex number z is z = reiθ , where eiθ is unimodular and r is positive, a complex matrix A can be represented as A = K ◦ ray(A), where K is an entry-wise positive matrix, and ◦ is the Hadamard product. We are interested in the problem of what properties of the complex matrix A can be determined just by ray(A). An n ×n ray pattern matrix A is said to require the weak (or strong) Perron–Frobenius property if for every n ×n entry-wise positive matrix K, K ◦A has the weak (or strong, respectively) Perron–Frobenius property. Then part of the Perron–Frobenius Theorem can be restated as: 0-1 square ray pattern matrices require the weak Perron–Frobenius prop-
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erty, and irreducible 0-1 square ray pattern matrices require the strong Perron–Frobenius property. Let Q(A) = {B| ray(B) = ray(A)}. Q(A) is called the ray pattern class of A. Then A requires the weak (strong) Perron–Frobenius property means for every matrix B ∈ Q(A), B has the weak (strong, respectively) Perron–Frobenius property. A natural problem arises: Question: Besides the 0-1 ray pattern matrices, are there any other ray pattern matrices that require the weak/strong Perron–Frobenius property? How does one characterize these ray pattern matrices? In [7], it is shown that if all of the cycles in the associated digraph of a square complex matrix A are negative, then A is diagonally similar to an SNS matrix. In this paper, we use a similar method to show that a ray pattern matrix requiring the weak Perron–Frobenius property if and only if it is diagonally similar to a 0-1 matrix; and that it requires the strong Perron–Frobenius property if and only if it is diagonally similar to an irreducible 0-1 matrix. 2. Main result Let A = (ajk )n×n be a complex square matrix with order n. Then A will sometimes be identified with its associated arc-weighted digraph WA = (D, w), where D is the digraph with the vertex set [n] = {1, 2, . . . , n}, for every j, k ∈ [n], (j, k) is an arc of D if and only if ajk = 0, and w((j, k)) = ajk . Note that there is no multi-arc in D, but it’s possible there are some loops in D. Loops are treated as directed cycles with length 1. (In this paper, all the cycles in digraph are understood to be directed cycles.) The cycles in WA are also called the cycles of A. Let C : j1 → j2 → . . . → js → j1 be a cycle of A, then C is called a positive cycle if the product aj1 j2 aj2 j3 · · · ajs j1 is positive; this product is also called the weight of the cycle C, and denoted by w(C). Lemma 2.1. Let A ∈ Mn . If the arc-weighted digraph of A is a cycle C with length s, and w(C) = w, then the spectrum of A consists of n − s copies of 0 and all s roots of the equation λs − w = 0. Proof. The conclusion follows from the facts that the characteristic polynomial of A is |λI − A| = λn − λn−s w = λn−s (λs − w). 2 To prove the necessity of our result, we need the following continuity theorem of the matrix spectrum with respect to its entries. Theorem 2.1. ([2, Theorem 2.4.9.2]) Let an infinite sequence A1 , A2 , . . . ∈ Mn be given, and suppose that limk→∞ Ak = A (entrywise convergence). Let λ(A) =
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[λ1 (A) . . . λn (A)]T and λ(Ak ) = [λ1 (Ak ) . . . λn (Ak )]T be given presentations of the eigenvalues of A and Ak , respectively, for k = 1, 2, . . .. Let Sn = {π : π is a permutation of {1, 2, . . . , n}} . Then for each given > 0 there exists a positive integer N = N () such that min
max
π∈Sn i=1,2,...,n
|λπ(i) (Ak ) − λi (A)| ≤ for all k ≥ N
Two square matrices A and B are said to be diagonally similar if there exists a diagonal invertible matrix Q = diag(q1 , q2 , . . . , qn ) such that Q−1 AQ = B. It is easy to see diagonal similarity is an equivalence relationship. Furthermore, from the view of graph theory, if WA and WB are the arc-weighted digraphs corresponding to A and B, respectively, then B = Q−1 AQ is equivalent to the statement that WB can be obtained from WA by multiplying the weights of all the arcs going out of vertex j by qj and multiplying the weights of all the arcs going into j by qj−1 for j = 1, . . . , n. We call this kind of graph operation an arc weight redistribution. The following lemma gives a necessary and sufficient condition for the diagonal similarity, it is a generalization of Theorem 4.1 in [7]. Also, the proof idea is similar. Lemma 2.2. Let A = (ajk )n×n , B = (bjk )n×n be complex irreducible square matrices, WA = (DA , wA ) and WB = (DB , wB ) be the arc-weighted digraphs corresponding to A and B, respectively. Then A is diagonally similar to B if and only if DA = DB , and for every cycle C in DA (and DB ), wA (C) = wB (C). Proof. Suppose that B = Q−1 AQ, where Q = diag(q1 , q2 , . . . , qn ). It is easy to see bjk = qj−1 ajk qk . Then ajk = 0 if and only if bjk = 0, thus DA = DB . Let C : j1 → j2 → · · · → js → j1 be a cycle in D. Then wB (C) = bj1 j2 bj2 j3 · · · bjs j1 = qj−1 aj1 j2 qj2 qj−1 aj2 j3 qj3 · · · qj−1 a js j1 q j1 1 2 s = wA (C). Now we use a method of graph theory to prove the sufficiency part. Note that A, B are irreducible, then DA (= DB ) are strongly connected. By the recursive constructions of strongly connected digraphs (see [1, P108]), we can assume that DA = D + P , where D is a strongly connected sub-digraph of DA , and P : x → k1 → k2 → · · · → kt → y is a directed path (or cycle) connecting two vertices x and y in D (if x = y, then P is a cycle). Without loss of generality, we may assume the vertex set of D is [m], where m is the order of D . Let WA (WB ) be the sub-digraph of WA (WB , respectively) whose underlying digraph is D .
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By induction on the number of arcs in DA , WB can be obtained by a proper arc weight redistribution from WA , now apply the same arc weight redistribution operation on the vertex set [m] of WA , and denote the result arc-weighted digraph by W = (DA , w). Then the only possible difference between WB and W are the weights of the arcs on the path P . By successively applying proper arc weight redistribution operations on the vertices k1 , k2 , . . . , kt , we can further assume that w((x, k1 )) = wB ((x, k1 )), w((k1 , k2 )) = wB ((k1 , k2 )), . . . , w((kt−1 , kt )) = wB ((kt−1 , kt )). For the last arc (kt , y), w(kt , y) has to be the same as wB (kt , y) since (kt , y) needs to belong to some cycle C in DA and w(C) = wB (C) for every cycle in DA . Then W = WB , which means WB can be obtained from WA by a suitable arc weight redistribution operation, yielding that A and B are diagonally similar. 2 Theorem 2.2. Let A be a square ray pattern matrix with order n. Then (a) A requires the weak Perron–Frobenius property if and only if the weight of every cycle in A is positive. (b) A requires the strong Perron–Frobenius property if and only if WA is strongly connected and the weight of every cycle in A is positive. Proof. (1) We first prove the necessity part. Suppose to the contrary that there exists a cycle C : j1 → j2 → · · · → js → j1 such that wA (C) = eiθ , where θ ∈ (0, 2π). Write K = K() = (kjs )n×n , where kjs = 1 if (j, s) ∈ C and kjs = otherwise. Write B = B() = K ◦ A. By Theorem 2.1 and Lemma 2.1, ρ(B) → 1 as → 0+ , but none of B’s eigenvalues will converge to 1. Thus for > 0 small enough, ρ(B) will not be an eigenvalue of B, contradicting the fact that A requires the Perron–Frobenius property. For the sufficiency part, after applying suitable simultaneous row and column permutations, without loss of generality we assume that A is a upper triangle block matrix whose diagonal blocks A1 , A2 , . . . Ar are all irreducible. For any entrywise positive matrix K, write B = K ◦ A, and B1 , B2 , . . . , Br to be the diagonal irreducible blocks of B. For each j = 1, 2, . . . , r, let Cj be the matrix obtained from Bj by replacing each of its nonzero entries with the corresponding modulus. Then the underlying digraph of Bj and Cj are the same. Since all the cycles in A are positive, then all the cycles in B1 , B2 , . . . , Br are also positive. Thus by Lemma 2.2, B1 , B2 , . . . , Br are diagonally similar to C1 , C2 , . . . , Cr , respectively. Since diagonally similar matrices share the same spectrum, then the ρ(Bj ) is an eigenvalue of Bj for every j = 1, 2, . . . , r by the Perron–Frobenius Theorem. Note that ρ(B) = maxj ρ(Bj ), and spectrum of B is the union of the spectra of B1 , B2 , . . . , Br , then ρ(B) is still an eigenvalue of B, thus A requires the weak Perron–Frobenius property. (2) It is well known that the irreducibility of A is equivalent to the strong connectivity of WA , so together with (1) and the Perron–Frobenius Theorem, the sufficiency part holds.
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For the necessity part, we only need to prove that A is irreducible. Suppose to the contrary, without loss of generality, assume that A is a upper triangle block matrix whose diagonal blocks A1 , A2 , . . . Ar are all irreducible. Let B = K ◦ A, B1 , B2 , . . . , Br be the diagonal blocks of B, respectively. By properly choosing K, we can assume that ρ(B) = ρ(B1 ) = ρ(B2 ), which means ρ(B) is not algebraically simple, contradicting to that A requires strong Perron–Frobenius property. 2 The Perron–Frobenius Theorem can be generalized from real field to complex field as below: Theorem 2.3. Let A ∈ Mn (not necessarily be a ray pattern matrix), n ≥ 2. If A is irreducible and all the cycles in A are positive, then (a) (b) (c) (d) (e)
ρ(A) > 0. ρ(A) is an algebraically simple eigenvalue of A. there is an entrywise nonzero vector x = [xj ] such that Ax = ρ(A)x. there is an entrywise nonzero vector y = [yj ] such that y T A = ρ(A)y T . xj yj are all positive for j = 1, 2, . . . , n.
Proof. Let B be the matrix obtained from A by replacing each of nonzero entries with the corresponding modulus. Then B is a irreducible nonnegative matrix. By Lemma 2.2, there exists an invertible matrix Q = diag(q1 , q2 , . . . , qn ) such that B = Q−1 AQ. By the Perron–Frobenius Theorem, corresponding to its eigenvalue ρ(B), B has an (B) entrywise positive right eigenvector x(B) = [xj ], and an entrywise positive left eigen(B)
vector y (B) = [yj
]. Choose x = Qx(B) , y = Q−1 y (B) , then x and y are the right and left
eigenvectors of A corresponding to ρ(A), respectively; and xj = qj xj , yj = qj−1 yj so xj yj is positive for every j = 1, 2, . . . , n, completing the proof. 2 (B)
(B)
,
Next we restrict our attention to ray pattern matrices. An n × n complex matrix S is defined to be a signature matrix if S is a diagonal matrix all of whose diagonal entries are unimodular. Note that S −1 = S¯ is still a signature matrix. An n × n complex matrix M is defined to be signature nonnegative (respectively, signature positive) if there exists an n × n signature matrix S such that SM S −1 is entrywise nonnegative (respectively, entrywise positive). We have the following two results. Theorem 2.4. (a) An n × n irreducible ray pattern matrix A is signature nonnegative if and only if all cycles in the arc-weighted directed graph WA are positive. (b) An n × n entrywise nonzero square ray pattern matrix A is signature positive if and only if the arc-weighted directed graph WA has all cycles be positive.
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Proof. The implication parts holds trivially for both of these results. We only need to prove the inverse implication parts. (1) Let B be the matrix obtained from A by replacing each of nonzero entries with 1. Since all the nonzero entries of A are unimodular, then for every cycle C in DA (= DB ), wA (C) = wB (C) = 1. By Lemma 2.2, there exists an invertible matrix S = diag(s1 , s2 , . . . , sn ) such that A = SBS −1 . Note that in the proof of Lemma 2.2, the arc weight redistribution operations do not change the modulus of the arc weights, thus s1 , s2 , . . . , sn are all unimodular, yielding that S is a signature matrix. The proof of (2) is similar. 2 Acknowledgements The authors would like to thank the referee for his/her careful reading, valuable suggestions and encouragement! Also, Theorem 2.4 is proposed by the referee. References [1] R.A. Brualdi, B.L. Shader, Matrices of Sign-Solvable Linear Systems, Vol. 116, Cambridge University Press, 1995. [2] R.A. Horn, C.R. Johnson, Matrix Analysis, second edition, Cambridge University Press, 2012. [3] C.R. Johnson, P. Tarazaga, On matrices with Perron–Frobenius properties and some negative entries, Positivity 8 (4) (2004) 327–338. [4] D. Noutsos, On Perron–Frobenius property of matrices having some negative entries, Linear Algebra Appl. 412 (2) (2006) 132–153. [5] P. Tarazaga, M. Raydan, A. Hurman, Perron–Frobenius theorem for matrices with some negative entries, Linear Algebra Appl. 328 (1) (2001) 57–68. [6] A. Elhashash, D.B. Szyld, On general matrices having the Perron–Frobenius property, Electron. J. Linear Algebra 17 (1) (2008) 389–413. [7] J.-Y. Shao, H.-Y. Shan, L.-H. You, Ray solvable linear systems and ray S 2 N S matrices, Linear Algebra Appl. 395 (2005) 229–246.