Geometric Mapping Properties of Semipositive Matrices

Transcription

1 Geometric Mapping Properties of Semipositive Matrices M. J. Tsatsomeros Mathematics Department Washington State University Pullman, WA July 14, 2015 Abstract Semipositive matrices map a positive vector to a positive vector and as such they are a very broad generalization of the irreducible nonnegative matrices. Nevertheless, the ensuing geometric mapping properties of semipositive matrices result in several parallels to the theory of cone preserving and cone mapping matrices. It is shown that for a semipositive matrix A, there exist a proper polyhedral cone K 1 of nonnegative vectors and a polyhedral cone K 2 of nonnegative vectors such that AK 1 = K 2. The set of all nonnegative vectors mapped by A to the nonnegative orthant is a proper polyhedral cone; as a consequence, A belongs to a proper polyhedral cone comprising semipositive matrices. When the powers A k (k = 0, 1,...) have a common semipositivity vector, then A has a positive eigenvalue. If A has a sole peripheral eigenvalue λ and the powers of A have a common semipositivity vector with a non-vanishing term in the direction of the left eigenspace of λ, then A leaves a proper cone invariant. Keywords: semipositive matrix, nonnegative matrix, Perron-Frobenius, proper cone, polyhedral cone, principal pivot transform, Cayley transform. AMS Subject Classifications: 15A48, 15A23, 15A18. Dedicated to Hans Schneider for his unwavering support for the linear algebra family. 1

2 1 Introduction By its definition, a semipositive matrix maps a positive vector to another positive vector. This is a trait shared by many important matrix classes, such as the nonnegative matrices, the positive definite matrices, the M-matrices and the P-matrices. The first systematic consideration of semipositive matrices occurs in [3], under the name class S. The basic properties of semipositive matrices, their subclasses and applications can be reviewed in [2, 3, 5]. It is enticing to study the characteristic property of semipositive matrices as a mapping property between geometric objects, specifically, convex cones in R n. Matrix semipositivity can indeed be viewed as the broadest possible generalization of the irreducible nonnegative matrices which map all positive vectors to positive vectors. As such, one would not expect that many strong properties of nonnegative matrices generalize to semipositive matrices. There are, however, some cone-theoretic and Perron-Frobeniustype implications of semipositivity, especially when all the powers of a matrix are assumed or implied to be semipositive. Our effort to reveal geometric and spectral consequences of semipositivity unfolds as follows: Section 2 contains general notation and definitions. Section 3 examines three factorizations of semipositive matrices into semipositive matrices with common semipositivity vectors. Section 4 contains some geometric observations regarding semipositive maps and examines the set of all nonnegative vectors that get mapped by a semipositive matrix to the nonnegative orthant. In Section 5, spectral results related to semipositivity are collected, in particular, as to when a semipositive matrix has a positive eigenvalue with a corresponding nonnegative eigenvector. 2 Notation and definitions The all-ones vector is denoted by e = [ ] T with its size determined by the context. The n n matrices with complex (resp., real) entries is denoted by M n (C) (resp., M n (R)). The following notation is used for A M n (C): The spectrum of A is σ(a), viewed as a multiset containing the eigenvalues according to their multiplicities. The spectral radius of A is ρ(a) = max{ λ : λ σ(a)}. Eigenvalues λ with λ = ρ(a) are referred to as peripheral. The index of λ σ(a) is the size of the largest Jordan block of λ in the Jordan canonical form of A and is denoted by ν λ. By convention, ν λ = 0 if and only if λ σ(a). For α {1, 2,..., n}, α = {1, 2,..., n} \ α. A[α, β] is the submatrix of A whose rows and columns are indexed by α, β {1, 2,..., n}, respectively; the elements of α, β are assumed to be in ascending order. When a row or column index set is empty, 2

3 the corresponding submatrix is considered vacuous and by convention has determinant equal to 1. A[α, α] is abbreviated by A[α]. We refer to an array X as nonnegative (resp., positive) if its entries are nonnegative (resp., positive) and we write X 0 (resp., X > 0). The nonnegative orthant in R n comprises all nonnegative vectors and is denoted by R n +; the topological interior of R n + comprises all positive vectors in R n and is denoted by int R n +. We use x > 0 and x int R n + interchangeably. Definition 2.1 Matrix A M n (R) is semipositive if there exists vector x 0 such that Ax > 0. Notice that by continuity of the map x Ax, semipositivity of A is equivalent to the existence of x > 0 such that Ax > 0; we refer to such an x > 0 as a semipositivity vector of A. Note the following two simple facts about semipositivity: If A M n (R) has a column all of whose entries are positive, then A is semipositive. If A has a row none of whose entries is positive, then A is not semipositive. Next, we recall the definitions of some matrix transforms. Given A M n (C) and α {1, 2,..., n} such that A[α] is invertible, A/A[α] denotes the Schur complement of A[α] in A, that is, A/A[α] = A[α] A[α, α]a[α] 1 A[α, α]. Definition 2.2 Given a nonempty α n and provided that A[α] is invertible, the principal pivot transform of A M n (C) relative to α is the matrix ppt (A, α) M n (C) obtained from A by replacing A[α] by A[α] 1, A[α, α] by A[α] 1 A[α, α], A[α, α] by A[α, α]a[α] 1, A[α] by A/A[α]. By convention, ppt (A, ) = A. To illustrate this definition, when α = {1, 2,..., k} for some positive integer k < n, we have that [ ] A[α] 1 A[α] 1 A[α, α] ppt (A, α) = A[α, α]a[α] 1. A/A[α] Observe that ppt (A, {1, 2,..., n}) = A 1. See [10] for the properties and applications of the principal pivot transform. Definition 2.3 Given A M n (C) with 1 σ(a), the Cayley transform of A is the fractional linear map F A = (I + A) 1 (I A). The map A F A is an involution, i.e., A = (I + F A ) 1 (I F A ). Note that as (I + A) 1 is a polynomial in A, it commutes with any other polynomial in A and so the order of the factors in the definition of F A is immaterial. 3

4 The following geometric concepts will be used in the sequel. A nonempty convex set K R n is said to be a cone if αk K for all α 0. A cone K is called proper if it is (i) closed (in the Euclidean space R n ), (ii) pointed (i.e., K ( K) = {0}), and (iii) solid (i.e., the topological interior of K, int K, is nonempty). A polyhedral cone K R n is a cone consisting of all nonnegative linear combinations of a finite set of vectors in R n, which are called the generators of K. Thus, K is polyhedral if and only if K = XR m + for some n m matrix X; when m = n and X is invertible, K = XR n + is called a simplicial cone in R n +. Note that simplicial cones in R n are proper cones. Given two proper cones K 1, K 2 in R n, we let L(K 1, K 2 ) denote the set of matrices A in M n (R) such that AK 1 K 2. The following concept is introduced in Schneider [8]. Definition 2.4 Given A M n (R) and a nonnegative integer k, the intrinsic cone, w k (A), of A is the cone consisting of all nonnegative linear combinations of the matrices A k, A k+1,... in M n (R). 3 Factorizations of semipositive matrices We begin with a factorization of a semipositive matrix into the product of a positive and an inverse positive matrix. Theorem 3.1 A M n (R) is semipositive if and only if there exist positive matrices X and Y such that X is invertible and A = Y X 1. Proof. Let A be semipositive, i.e., there exist positive vectors x, y R n such that Ax = y. Define the n n matrices X = xe T + ɛi, Y = ye T + ɛa, where ɛ > 0 is sufficiently small to have Y > 0. Then the result follows from the facts that AX = Y and that X > 0 is invertible since its eigenvalues are ɛ and e T x + ɛ. For the converse, assume there are positive matrices X and Y such that A = Y X 1. Let u > 0 and set v = Xu > 0. Then Av = Y X 1 v = Y u > 0, completing the proof of the theorem. We continue with two more factorizations of semipositive matrices based on the principal pivot transform (Theorem 3.4) and the Cayley transform (Theorem 3.6). Theorem 3.2 Let A M n (R) and α {1, 2,..., n} such that A[α] is invertible. Then A is semipositive if and only if the principal pivot transform ppt (A, α) is semipositive. 4

5 Proof. Let B = ppt (A, α) and suppose A is semipositive, i.e., y = Ax > 0 for some x > 0. Define u, v R n such that u[α] = y[α], u[α] = x[α] and v[α] = x[α], v[α] = y[α] By the nature of the principal pivot transform [10], we have that Ax = y if and only if Bu = v. As x and y are assumed to be positive, it follows that u > 0, v > 0 and so B is semipositive. The converse follows from the fact that the principal pivot transform is an involution, that is, A = ppt (B, α). The following lemma (see [10, Lemma 3.4]), provides a factorization of the principal pivot transform. Lemma 3.3 Let B M n (R) and α n so that B[α] is invertible. Let T 1 be the matrix obtained from the identity by setting the diagonal entries indexed by α equal to 0. Let T 2 = I T 1 and consider the matrices C 1 = T 2 + T 1 B, C 2 = T 1 + T 2 B. Then ppt (B, α) = C 1 C 1 2. Theorem 3.4 Let A M n (R) and nonempty α {1, 2,..., n} such that B = ppt (A, α) is well-defined. Let ppt (B, α) = C 1 C2 1 be the factorization in Lemma 3.3. Then A = C 1 C2 1 is semipositive if and only if C 1 and C 2 are semipositive with a common semipositivity vector. Proof. Without loss of generality, assume that α = {1, 2,..., k}; otherwise, our arguments apply to a permutation similarity of A. Let B = ppt (A. α) and observe that [ ] [ ] [ ] B[α] B[α, α] I 0 B[α] B[α, α] B =, C 1 = and C 2 =. B[α, α] B[α] B[α, α] B[α] 0 I Since the principal pivot transform is an involution, we have A = ppt (B, α) = C 1 C 1 2. If A is semipositive, then by Theorem 3.2, B is semipositive. By the definition of C 1 and C 2, for any semipositivity vector u of B we have C 1 u > 0 and C 2 u > 0. Conversely, observe that any common semipositivity vector of C 1 and C 2 is a semipositivity vector of B and thus a semipositivity vector of A by Theorem 3.2. Example 3.5 Let A = , B = ppt (A, {1, 2}) =

6 By Theorem 3.2, the matrix A is semipositive because its principal pivot transform B has a positive column and is thus semipositive. For example, Bu > 0 for u = [4 1 1] T satisfies Bu > 0. According to Theorem 3.4, A admits the following factorization into semipositive matrices (with common semipositivity vector u): A = Theorem 3.6 Let A M n (R) so that 0 and 1 are not eigenvalues of A. Then the Cayley transform F = F A of A is well-defined and A = (I F )(I + F ) 1 is semipositive if and only if I F and I + F are semipositive with a common semipositivity vector. Proof. Since 0 and 1 are not eigenvalues of A, A is invertible and F = F A is welldefined. Furthermore, by [1, Lemma 2.2], we have that (I + F ) 1 = 1 2 (I + A) and (I F ) = 2(I + A 1 ) 1. (3.1) Assume first that A is semipositive, i.e., y = Ax > 0 for some x > 0. Then by (3.1), 1 (I + F ) 1 x = 1 2 (I + A) x = 1 (x + y) > 0 2. and That is, as (I F ) x + y 2 = 2(I + A 1 ) 1 x + y 2 = y > 0. (I + F )(x + y) = 2x > 0 and (I F )(x + y) = 2y > 0, we have that I +F and I F are semipositive with common semipositivity vector x+y. Conversely, suppose that I + F and I F are semipositive with common semipositivity vector u > 0, i.e., (I + F )u = v > 0 and (I F )u > 0. By (3.1) we obtain A = 2(I + F ) 1 I and so Av = 2(I + F ) 1 v v = 2u v. However, 2u v = u + u v = v F u + u v = (I F )u > 0; i.e., A is semipositive. 4 Geometric properties of semipositive matrices Each of the three factorizations of a semipositive matrix A provided in Section 3 is of the form Y X 1, where X and Y are semipositive with a common semipositivity vector. As such, each of these factorizations leads to the existence of polyhedral cones K 1 = XR n + and K 2 = Y R n + such that AK 1 = K 2, where K 1 K 2 contains a positive vector. We state and prove this formally for the factorization in Theorem

7 Theorem 4.1 A M n (R) is semipositive if and only there exist proper polyhedral cone K 1 R n + and polyhedral cone K 2 int R n + {0} such that AK 1 = K 2. Proof. Let A be semipositive. Consider the matrices X, Y in the proof of Theorem 3.1 such that A = Y X 1 and let K 1 = X R n + and K 2 = Y R n +. Since X is positive and invertible, K 1 is a simplicial and thus a proper cone in R n +. Since Y > 0, K 2 is a polyhedral cone in int R n + {0}. We also have that AK 1 = Y X 1 XR + = K 2. For the converse, suppose there exist a proper cone K 1 R n + and a polyhedral cone K 2 int R n + {0} such that AK 1 = K 2. As K 1 R n + is proper, it is solid and so there is x intk 1 (i.e., x > 0 ). It follows that Ax K 2 \ {0} (i.e., Ax > 0). Thus A is semipositive. Example 4.2 Let A = , which is semipositive since y = Ax = [5 3 1] T for x = [2 3 4] T. Then, by Theorem 3.1, A = Y X 1, where X = xe T + I = and Y = ye T + A = By Theorem 4.1, AK 1 = K 2, where K 1 = XR n + R n +, K 2 = Y R n + R n +. Note that as we have that x K 1 and y K 2. X x = (e T x + 1) x and Y x = (e x + 1) y, As seen in the proof of Theorem 4.1, a vector x > 0 naturally ties together all semipositive matrices that have x as their semipositivity vector. As a consequence, we are led to study the set of nonnegative vectors that get mapped into the nonnegative orthant by a given semipositive matrix. Definition 4.3 Let A M n (R) be semipositive. The semipositive cone of A is K A = {x 0 : Ax 0}. As its name implies, K A is a convex cone in R n that contains all the semipositivity vectors of A. More can be said about this cone. 7

8 Theorem 4.4 Let A M n (R) be semipositive. Then K A is a proper polyhedral cone in R n. Proof. K A is clearly a convex set that is closed under nonnegative scaling and is nonempty, since A is semipositive; that is, K A is a cone in R n. Let x > 0 such that Ax > 0 and let X, Y, K 1, K 2 as in the proof of Theorem 4.1, so that AK 1 = K 2. Hence, K 1 is a simplicial and consequently a proper polyhedral cone that is contained in K A. It follows that K A contains a solid cone and therefore K A is solid. Also, K A R n + and consequently K is pointed. Lastly, the limit points of K A lie in K A and so K A is closed. Thus, K A is proper cone in R n. Notice next that K A is the intersection of two polyhedral cones, specifically, R n + and the cone generated by the columns of A. A result of Minkowski and Weyl (see [6, Theorem 12, Chapter XVIII]) implies that K A is itself a polyhedral cone, completing the proof of the theorem. Next we observe that a semipositive matrix A belongs to a proper polyhedral cone in M n (R) defined via K A. Theorem 4.5 Let A M n (R) be semipositive. Then L(K A, R n +) is a proper polyhedral cone in M n (R) and A L(K A, R n +). Proof. It is clear by the definition of the semipositive cone of A that AK A R n +, i.e., A L(K A, R n +). Moreover, by Theorem 4.4, K A is a proper polyhedral cone in R n, and so is R n +. It follows by Schneider and Vidyasagar [9, Lemma 9] that L(K A, R n +) is a proper polyhedral cone in M n (R). Remark 4.6 Note that if A is semipositive and B L(K A, R n +), then for every u K A, Bu 0. Thus K A K B. 5 Spectral considerations Semipositivity, by itself, does not impose any spectral restrictions on a matrix at all; that is, any real polynomial can be the characteristic polynomial of a semipositive matrix. This is formalized in the next result. Proposition 5.1 Given the spectrum σ of any real n n matrix, there exists a semipositive matrix A such that σ(a) = σ. Proof. By the results in [7], an n n sign pattern A that consists of some positive and some negative columns, with at least one of each kind, is spectrally arbitrary. This 8

9 means that the signed entries of A can be chosen so that the resulting matrix A has any given self-conjugate multiset with n members as its spectrum. Such an A has a column all of whose entries are positive and so it is semipositive. For special subclasses of semipositive matrices, there are indeed spectral impositions: When A is a nonnegative irreducible matrix, the Perron-Frobenius theorem imposes that ρ(a) is a simple eigenvalue corresponding to a positive eigenvector. The right-hand most eigenvalue of a nonsingular M-matrix is positive and corresponds to a positive eigenvector, while all eigenvalues have positive real parts. The eigenvalues of a positive definite matrix are all positive numbers. The eigenvalues of an n n P-matrix can not lie in the open sector {z C : π π n < Arg z < π + π }. Interestingly, most of the above considerations extend to matrix powers: The powers of an irreducible nonnegative matrix n are indeed semipositive with any positive vector serving as a common semipositivity vector. The powers of an irreducible M-matrix A = si B (B > 0, s > ρ(b)) are also semipositive with the Perron vector x of B acting as a common semipositivity vector. In addition, the nonnegative matrices, the positive definite matrices and the totally positive matrices are closed under the taking of powers. As a consequence, it is worth considering the case of matrices all of whose powers are semipositive. Theorem 5.2 Let A M n (R) whose index of the eigenvalue 0 is ν 0. Suppose that the powers A k for k ν 0 have a common semipositivity vector. Then A has a positive eigenvalue. Proof. Let k be a nonnegative integer such that k ν 0. By Schneider [8, Theorem 1.6], A M n (R) has a positive eigenvalue if and only if the intrinsic cone, w k (A), is a nonzero pointed cone. By way of contradiction, if w k (A) is not pointed, then there exists nonzero matrix B such that B and B belongs to w k (A). By assumption, there also exists x > 0 such that A k x > 0 for all k ν 0. Thus Bx > 0 and Bx > 0; a contradiction that shows that w k (A) is pointed and thus A has a positive eigenvalue. Following is a direct consequence of Theorem 5.2, which can also be obtained directly from [8, Theorem 6.4] since a matrix whose powers are semipositive is necessarily nonnilpotent. Corollary 5.3 Let A M n (R) and suppose that the powers A k for all k 1 have a common semipositivity vector. Then A has a positive eigenvalue. If A has a positive eigenvalue corresponding to a positive eigenvector, then clearly the powers A k are semipositive with a common semipositivity vector. A weaker converse is stated next, where R( ) and N( ) denote the range and nullspace of a matrix. Theorem 5.4 Let A M n (R) so that λ C is a simple eigenvalue of A and λ > µ for all eigenvalues µ λ of A. If the powers A k are semipositive for all integers k 1 9

10 and have a common semipositivity vector x R(A λi), then λ > 0 and there exists a nonnegative eigenvector of A corresponding to λ. Proof. Let u, v be right and left eigenvectors of A corresponding to λ, respectively, normalized so that v T u = 1. By assumption, u and v span the eigenspaces of A and A T corresponding to λ 0, respectively, and ρ(a) = λ > µ for every eigenvalue µ λ of A. Thus by [4, Lemma 8.2.7] we have that ( ) A k lim = uv T. (5.2) k ρ(a) Suppose now that A k is semipositive for all integers k 1 with common semipositivity vector x such that x R(A λi). By (5.2) and continuity uv T x = (v T x)u 0. Since x R(A λi) = N(A T λi), we have that v T x 0; that is, u or u is a nonnegative eigenvector of A corresponding to λ. It remains to show that λ > 0. The dominance assumptions for λ imply that λ must be nonzero and real; otherwise λ λ and λ = λ. If it were that λ < 0, then the sequence ( ) A k ( ) λ k x = x ρ(a) ρ(a) would not converge to uv T x as k, contradicting (5.2) that it should converge to the nonzero, nonnegative or nonpositive vector uv T x. Thus λ > 0, concluding the proof of the theorem. Note that the condition that x R(A λi) in Theorem 5.4, is equivalent to stating that in the representation of x as a sum of generalized eigenvectors corresponding to the distinct eigenvalues of A T, there is a non-vanishing term in the direction of the eigenspace of λ. We can now state a connection to the theory of cone invariant maps. Corollary 5.5 Let A M n (R) so that λ C is a simple eigenvalue of A and λ > µ for all eigenvalues µ λ of A. If the powers A k are semipositive for all integers k 1 and have a common semipositivity vector x R(A λi), then there exists a proper cone K in R n such that AK K. Proof. Let A and λ as prescribed and suppose that A k is semipositive for all integers k 1. By Theorem 5.4, the dominant eigenvalue λ is equal to the spectral radius of A. Furthermore, the index of any other peripheral eigenvalue of A is less than the index of λ; this holds by default because by assumption there are not any eigenvalues of A, other than λ, of modulus ρ(a). Thus the Perron-Schaefer condition holds for A and the existence of the proper cone K left invariant by A follows from Vandergraft [11, 3, Theorem 3.5]. 10

11 Remark 5.6 Matrix A M n (R) is called minimally semipositive if it is semipositive and if the deletion of any of the columns of A results in an n (n 1) matrix that does not map any positive vector in R n 1 to a positive vector in R n +. For example, M-matrices are minimally semipositive because deletion of any of their columns results in a matrix with a row all of whose entries are non-positive. It is well-known that A M n (R) is minimally semipositive if and only if A is invertible and A 1 0; see [5, Theorem 3.4]. As a consequence, for a minimally semipositive A M n (R) we have the following conclusions related to our discussion: (i) A 1 R n + R n +; (ii) by the Perron-Frobenius theorem applied to A 1, there exists nonzero x 0 such that Ax = τ(a)x, where τ(a) = min{ λ : λ σ(a)} > 0; (iii) the powers A k (k = 1, 2,...) are semipositive with a common semipositivity vector. References [1] S.M. Fallat and M.J. Tsatsomeros. On the Cayley transform of positivity classes of matrices. Electronic Journal of Linear Algebra, 9: , [2] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia, [3] M. Fiedler and V. Pták. Some generalizations of positive definiteness and monotonicity. Numerische Mathematik, 9: , [4] R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, [5] C.R. Johnson, M. K Kerr, and D.P. Stanford. Semipositivity of Matrices. Linear and Multilinear Algebra, 37: , [6] M. Gerstenhaber. Theory of Convex Polyhedral Cones, in Activities Analysis of Production and Allocation (T.C. Koopmans, Ed.), Wiley, New York, [7] J.J. McDonald, D.D. Olesky, M.J. Tsatsomeros, and P. van den Driessche. On the spectra of striped sign patterns. Linear and Multilinear Algebra, 51:39-48, [8] H. Schneider. Geometric conditions for the existence of positive eigenvalues of matrices. Linear Algebra and its Applications, 38: , [9] H. Schneider and M. Vidyasagar. Cross-positive matrices. SIAM Journal of Numerical Analysis, 7(4): , [10] M. Tsatsomeros. Principal pivot transforms: properties and applications. Linear Algebra and Its Applications, 300: , [11] J.S. Vandergraft. Spectral properties of matrices which have invariant cones. SIAM Journal of Applied Mathematics, 16: ,