Integral Equations and Operator Theory, Mar 1, 2005
Let K 1 , K 2 be closed, full, pointed convex cones in finite-dimensional real vector spaces of t... more Let K 1 , K 2 be closed, full, pointed convex cones in finite-dimensional real vector spaces of the same dimension, and let F : K 1 → span K 2 be a homogeneous, continuous, K 2-convex map that satisfies F (∂K 1) ∩ int K 2 = ∅ and F K 1 ∩ int K 2 = ∅. Using an equivalent formulation of the Borsuk-Ulam theorem in algebraic topology, we show that we have F (K 1 \{0}) ∩ (−K 2) = ∅ and K 2 ⊆ F K 1. We also prove that if, in addition, G : K 1 → span K 2 is any homogeneous, continuous map which is (K 1 , K 2)-positive and K 2-concave, then there exist a unique real scalar ω 0 and a (up to scalar multiples) unique nonzero vector x 0 ∈ K 1 such that Gx 0 = ω 0 F x 0 , and moreover we have ω 0 > 0 and x 0 ∈ int K 1 and we also have a characterization of the scalar ω 0. Then, we reformulate the above result in the setting when K 1 is replaced by a compact convex set and recapture a classical result of Ky Fan on the equilibrium value of a finite system of convex and concave functions.
By the signless Laplacian of a (simple) graph G we mean the matrix Q (G) = D(G) + A(G), where A(G... more By the signless Laplacian of a (simple) graph G we mean the matrix Q (G) = D(G) + A(G), where A(G), D(G) denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of G. For every pair of positive integers n, k, it is proved that if 3 ≤ k ≤ n − 3, then H n,k , the graph obtained from the star K 1,n−1 by joining a vertex of degree 1 to k+1 other vertices of degree 1, is the unique connected graph that maximizes the largest signless Laplacian eigenvalue over all connected graphs with n vertices and n + k edges.
Let K be a full, pointed closed cone in a finite dimensional real vector space. For any linear ma... more Let K be a full, pointed closed cone in a finite dimensional real vector space. For any linear map A for which AK⊆K, denote by (E,P(A)[E,I (A))] the directed graph whose vertex set consists of all the extreme rays of K such that there is an edge from F to G iff . It is proved that K is a
An equivalent condition on a 3-square complex or a 4-square real upper triangular matrix is found... more An equivalent condition on a 3-square complex or a 4-square real upper triangular matrix is found for its numerical range to be a circular disk centered at the origin. Sufficient conditions for the circularity of the numerical range of certain sparse matrices are also given in terms of graphs. Let A be an n-squ set denoted and define e nu range 0 was supported in p 0 Elsevier Science
Journal of Mathematical Analysis and Applications, May 1, 2010
Let K be a proper (i.e., closed, pointed, full convex) cone in R n. An n × n matrix A is said to ... more Let K be a proper (i.e., closed, pointed, full convex) cone in R n. An n × n matrix A is said to be K-primitive if there exists a positive integer k such that A k (K \ {0}) ⊆ int K ; the least such k is referred to as the exponent of A and is denoted by γ (A). For a polyhedral cone K , the maximum value of γ (A), taken over all K-primitive matrices A, is called the exponent of K and is denoted by γ (K). It is proved that if K is an n-dimensional polyhedral cone with m extreme rays then for any K-primitive matrix A, γ (A) (m A − 1)(m − 1) + 1, where m A denotes the degree of the minimal polynomial of A, and the equality holds only if the digraph (E, P(A, K)) associated with A (as a cone-preserving map) is equal to the unique (up to isomorphism) usual digraph associated with an m × m primitive matrix whose exponent attains Wielandt's classical sharp bound. As a consequence, for any ndimensional polyhedral cone K with m extreme rays, γ (K) (n − 1)(m − 1) + 1. Our work answers in the affirmative a conjecture posed by Steve Kirkland about an upper bound of γ (K) for a polyhedral cone K with a given number of extreme rays.
Let K be a proper cone in R n , let A be an n×n real matrix that satisfies AK ⊆ K, let b be a giv... more Let K be a proper cone in R n , let A be an n×n real matrix that satisfies AK ⊆ K, let b be a given vector of K, and let λ be a given positive real number. The following two linear equations are considered in this paper: (i) (λI n − A)x = b, x ∈ K, and (ii) (A − λI n)x = b, x ∈ K. We obtain several equivalent conditions for the solvability of the first equation. For the second equation we give an equivalent condition for its solvability in case when λ > ρ b (A), and we also find a necessary condition when λ = ρ b (A) and also when λ < ρ b (A), sufficiently close to ρ b (A), where ρ b (A) denotes the local spectral radius of A at b. With λ fixed, we also consider the questions of when the set (A − λI n)K K equals {0} or K, and what the face of K generated by the set is. Then we derive some new results about local spectral radii and Collatz-Wielandt sets (or numbers) associated with a cone-preserving map, and extend a known characterization of M-matrices among Z-matrices in terms of alternating sequences.
A connected graph is said to be unoriented Laplacian maximizing if the spectral radius of its uno... more A connected graph is said to be unoriented Laplacian maximizing if the spectral radius of its unoriented Laplacian matrix attains the maximum among all connected graphs with the same number of vertices and the same number of edges. A graph is said to be threshold (maximal) if its degree sequence is not majorized by the degree sequence of any other graph (and, in addition, the graph is connected). It is proved that an unoriented Laplacian maximizing graph is maximal and also that there are precisely two unoriented Laplacian maximizing graphs of a given order and with nullity 3. Our treatment depends on the following known characterization: a graph G is threshold (maximal) if and only if for every pair of vertices u, v of G, the sets N (u)\{v}, N (v)\{u}, where N(u) denotes the neighbor set of u in G, are comparable with respect to the inclusion relation (and, in addition, the graph is connected). A conjecture about graphs that maximize the unoriented Laplacian matrix among all graphs with the same number of vertices and the same number of edges is also posed.
Transactions of the American Mathematical Society, Jul 12, 2000
For an n × n nonnegative matrix P , an isomorphism is obtained between the lattice of initial sub... more For an n × n nonnegative matrix P , an isomorphism is obtained between the lattice of initial subsets (of {1, • • • , n}) for P and the lattice of P-invariant faces of the nonnegative orthant R n +. Motivated by this isomorphism, we generalize some of the known combinatorial spectral results on a nonnegative matrix that are given in terms of its classes to results for a conepreserving map on a polyhedral cone, formulated in terms of its invariant faces. In particular, we obtain the following extension of the famous Rothblum index theorem for a nonnegative matrix: If A leaves invariant a polyhedral cone K, then for each distinguished eigenvalue λ of A for K, there is a chain of m λ distinct A-invariant join-irreducible faces of K, each containing in its relative interior a generalized eigenvector of A corresponding to λ (referred to as semidistinguished A-invariant faces associated with λ), where m λ is the maximal order of distinguished generalized eigenvectors of A corresponding to λ, but there is no such chain with more than m λ members. We introduce the important new concepts of semi-distinguished A-invariant faces, and of spectral pairs of faces associated with a cone-preserving map, and obtain several properties of a cone-preserving map that mostly involve these two concepts, when the underlying cone is polyhedral, perfect, or strictly convex and/or smooth, or is the cone of all real polynomials of degree not exceeding n that are nonnegative on a closed interval. Plentiful illustrative examples are provided. Some open problems are posed at the end.
Let K be a proper cone in R n , let A be an n×n real matrix that satisfies AK ⊆ K, let b be a giv... more Let K be a proper cone in R n , let A be an n×n real matrix that satisfies AK ⊆ K, let b be a given vector of K, and let λ be a given positive real number. The following two linear equations are considered in this paper: (i) (λI n − A)x = b, x ∈ K, and (ii) (A − λI n)x = b, x ∈ K. We obtain several equivalent conditions for the solvability of the first equation. For the second equation we give an equivalent condition for its solvability in case when λ > ρ b (A), and we also find a necessary condition when λ = ρ b (A) and also when λ < ρ b (A), sufficiently close to ρ b (A), where ρ b (A) denotes the local spectral radius of A at b. With λ fixed, we also consider the questions of when the set (A − λI n)K K equals {0} or K, and what the face of K generated by the set is. Then we derive some new results about local spectral radii and Collatz-Wielandt sets (or numbers) associated with a cone-preserving map, and extend a known characterization of M-matrices among Z-matrices in terms of alternating sequences.
Let K,, KS he cones. We say that K, is a subcone of Kz if EktK, cJW Kp Furthermore, if K1 # K2, K... more Let K,, KS he cones. We say that K, is a subcone of Kz if EktK, cJW Kp Furthermore, if K1 # K2, K1 is called a proper subcone; if dimK, =dim K2, K, is called a non-degenerate s&cone. We first prove that every n-dimensional indecomposable cone, n > 3, contains a non-degenerate indecomposable subcone which has no more than Zn-2 extremals. Then we construct for each n > 3 an n-dimensional indecomposable cone with exactly 2n-2 extremals such that each of its proper non-degenerate subcones is decomposable. *The contents of this paper formed part of the author's Ph.D. thesis at the University of Hong Kong, 1977. Thanks are due to Dr. Y. H. Au-Yeung for his advice and encouragement. 'After he had submitted the paper for publication, the author learned that a similar notion was also considered by Fiedler and P&k (see [3]).
J.H. Drew et al. [Linear and Multilinear Algebra 37 (1994) 304] conjectured that for n 4, the com... more J.H. Drew et al. [Linear and Multilinear Algebra 37 (1994) 304] conjectured that for n 4, the completely positive (CP) rank of every n × n completely positive matrix is at most [n 2 /4]. In this paper we prove that the CP rank of a 5 × 5 completely positive matrix which has at least one zero entry is at most 6, thus providing new supporting evidence for the conjecture.
We offer an almost self-contained development of Perron-Frobenius type results for the numerical ... more We offer an almost self-contained development of Perron-Frobenius type results for the numerical range of an (irreducible) nonnegative matrix, rederiving and completing the previous work of Issos, Nylen and Tam, and Tam and Yang on this topic. We solve the open problem of characterizing nonnegative matrices whose numerical ranges are regular convex polygons with center at the origin. Some related results are obtained and some open problems are also posed.
For every integer n≥4, it is proved that there is a unique graph of order n which maximizes the s... more For every integer n≥4, it is proved that there is a unique graph of order n which maximizes the spectral radius of the unoriented Laplacian matrix over all bicyclic graphs of order n, namely, the graph obtained from the cycle C4 by first adding a chord and then attaching n − 4 pendant edges to one end of the chord.
The purpose of this paper is to characterize those linear transformations on the space of $n \tim... more The purpose of this paper is to characterize those linear transformations on the space of $n \times n$ real matrices which map the class of $n \times n$ inverse $M$- matrices (or, the closure of this class) onto itself. As a by-product of our approach, we also obtain a sufficient condition for an inverse $M$-matrix (resp. $M$-matrix) to have all positive powers being inverse $M$-matrices (resp. $M$-matrices).
Cone-preserving map K-primitive matrix Exponents Polyhedral cone Exp-maximal cone Exp-maximal K-p... more Cone-preserving map K-primitive matrix Exponents Polyhedral cone Exp-maximal cone Exp-maximal K-primitive matrix Cone-equivalence Minimal cone Let K be a proper (i.e., closed, pointed, full convex) cone in R n. An n × n matrix A is said to be K-primitive if there exists a positive integer k such that A k (K \ {0}) ⊆ int K; the least such k is referred to as the exponent of A and is denoted by γ (A). For a polyhedral cone K, the maximum value of γ (A), taken over all K-primitive matrices A, is called the exponent of K and is denoted by γ (K). It is proved that the maximum value of γ (K) as K runs through all ndimensional minimal cones (i.e., cones having n + 1 extreme rays) is n 2 − n + 1 if n is odd, and is n 2 − n if n is even, the maximum value of the exponent being attained by a minimal cone with a balanced relation for its extreme vectors. The K-primitive matrices A such that γ (A) attain the maximum value are identified up to cone-equivalence modulo positive scalar multiplication.
Integral Equations and Operator Theory, Mar 1, 2005
Let K 1 , K 2 be closed, full, pointed convex cones in finite-dimensional real vector spaces of t... more Let K 1 , K 2 be closed, full, pointed convex cones in finite-dimensional real vector spaces of the same dimension, and let F : K 1 → span K 2 be a homogeneous, continuous, K 2-convex map that satisfies F (∂K 1) ∩ int K 2 = ∅ and F K 1 ∩ int K 2 = ∅. Using an equivalent formulation of the Borsuk-Ulam theorem in algebraic topology, we show that we have F (K 1 \{0}) ∩ (−K 2) = ∅ and K 2 ⊆ F K 1. We also prove that if, in addition, G : K 1 → span K 2 is any homogeneous, continuous map which is (K 1 , K 2)-positive and K 2-concave, then there exist a unique real scalar ω 0 and a (up to scalar multiples) unique nonzero vector x 0 ∈ K 1 such that Gx 0 = ω 0 F x 0 , and moreover we have ω 0 > 0 and x 0 ∈ int K 1 and we also have a characterization of the scalar ω 0. Then, we reformulate the above result in the setting when K 1 is replaced by a compact convex set and recapture a classical result of Ky Fan on the equilibrium value of a finite system of convex and concave functions.
By the signless Laplacian of a (simple) graph G we mean the matrix Q (G) = D(G) + A(G), where A(G... more By the signless Laplacian of a (simple) graph G we mean the matrix Q (G) = D(G) + A(G), where A(G), D(G) denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of G. For every pair of positive integers n, k, it is proved that if 3 ≤ k ≤ n − 3, then H n,k , the graph obtained from the star K 1,n−1 by joining a vertex of degree 1 to k+1 other vertices of degree 1, is the unique connected graph that maximizes the largest signless Laplacian eigenvalue over all connected graphs with n vertices and n + k edges.
Let K be a full, pointed closed cone in a finite dimensional real vector space. For any linear ma... more Let K be a full, pointed closed cone in a finite dimensional real vector space. For any linear map A for which AK⊆K, denote by (E,P(A)[E,I (A))] the directed graph whose vertex set consists of all the extreme rays of K such that there is an edge from F to G iff . It is proved that K is a
An equivalent condition on a 3-square complex or a 4-square real upper triangular matrix is found... more An equivalent condition on a 3-square complex or a 4-square real upper triangular matrix is found for its numerical range to be a circular disk centered at the origin. Sufficient conditions for the circularity of the numerical range of certain sparse matrices are also given in terms of graphs. Let A be an n-squ set denoted and define e nu range 0 was supported in p 0 Elsevier Science
Journal of Mathematical Analysis and Applications, May 1, 2010
Let K be a proper (i.e., closed, pointed, full convex) cone in R n. An n × n matrix A is said to ... more Let K be a proper (i.e., closed, pointed, full convex) cone in R n. An n × n matrix A is said to be K-primitive if there exists a positive integer k such that A k (K \ {0}) ⊆ int K ; the least such k is referred to as the exponent of A and is denoted by γ (A). For a polyhedral cone K , the maximum value of γ (A), taken over all K-primitive matrices A, is called the exponent of K and is denoted by γ (K). It is proved that if K is an n-dimensional polyhedral cone with m extreme rays then for any K-primitive matrix A, γ (A) (m A − 1)(m − 1) + 1, where m A denotes the degree of the minimal polynomial of A, and the equality holds only if the digraph (E, P(A, K)) associated with A (as a cone-preserving map) is equal to the unique (up to isomorphism) usual digraph associated with an m × m primitive matrix whose exponent attains Wielandt's classical sharp bound. As a consequence, for any ndimensional polyhedral cone K with m extreme rays, γ (K) (n − 1)(m − 1) + 1. Our work answers in the affirmative a conjecture posed by Steve Kirkland about an upper bound of γ (K) for a polyhedral cone K with a given number of extreme rays.
Let K be a proper cone in R n , let A be an n×n real matrix that satisfies AK ⊆ K, let b be a giv... more Let K be a proper cone in R n , let A be an n×n real matrix that satisfies AK ⊆ K, let b be a given vector of K, and let λ be a given positive real number. The following two linear equations are considered in this paper: (i) (λI n − A)x = b, x ∈ K, and (ii) (A − λI n)x = b, x ∈ K. We obtain several equivalent conditions for the solvability of the first equation. For the second equation we give an equivalent condition for its solvability in case when λ > ρ b (A), and we also find a necessary condition when λ = ρ b (A) and also when λ < ρ b (A), sufficiently close to ρ b (A), where ρ b (A) denotes the local spectral radius of A at b. With λ fixed, we also consider the questions of when the set (A − λI n)K K equals {0} or K, and what the face of K generated by the set is. Then we derive some new results about local spectral radii and Collatz-Wielandt sets (or numbers) associated with a cone-preserving map, and extend a known characterization of M-matrices among Z-matrices in terms of alternating sequences.
A connected graph is said to be unoriented Laplacian maximizing if the spectral radius of its uno... more A connected graph is said to be unoriented Laplacian maximizing if the spectral radius of its unoriented Laplacian matrix attains the maximum among all connected graphs with the same number of vertices and the same number of edges. A graph is said to be threshold (maximal) if its degree sequence is not majorized by the degree sequence of any other graph (and, in addition, the graph is connected). It is proved that an unoriented Laplacian maximizing graph is maximal and also that there are precisely two unoriented Laplacian maximizing graphs of a given order and with nullity 3. Our treatment depends on the following known characterization: a graph G is threshold (maximal) if and only if for every pair of vertices u, v of G, the sets N (u)\{v}, N (v)\{u}, where N(u) denotes the neighbor set of u in G, are comparable with respect to the inclusion relation (and, in addition, the graph is connected). A conjecture about graphs that maximize the unoriented Laplacian matrix among all graphs with the same number of vertices and the same number of edges is also posed.
Transactions of the American Mathematical Society, Jul 12, 2000
For an n × n nonnegative matrix P , an isomorphism is obtained between the lattice of initial sub... more For an n × n nonnegative matrix P , an isomorphism is obtained between the lattice of initial subsets (of {1, • • • , n}) for P and the lattice of P-invariant faces of the nonnegative orthant R n +. Motivated by this isomorphism, we generalize some of the known combinatorial spectral results on a nonnegative matrix that are given in terms of its classes to results for a conepreserving map on a polyhedral cone, formulated in terms of its invariant faces. In particular, we obtain the following extension of the famous Rothblum index theorem for a nonnegative matrix: If A leaves invariant a polyhedral cone K, then for each distinguished eigenvalue λ of A for K, there is a chain of m λ distinct A-invariant join-irreducible faces of K, each containing in its relative interior a generalized eigenvector of A corresponding to λ (referred to as semidistinguished A-invariant faces associated with λ), where m λ is the maximal order of distinguished generalized eigenvectors of A corresponding to λ, but there is no such chain with more than m λ members. We introduce the important new concepts of semi-distinguished A-invariant faces, and of spectral pairs of faces associated with a cone-preserving map, and obtain several properties of a cone-preserving map that mostly involve these two concepts, when the underlying cone is polyhedral, perfect, or strictly convex and/or smooth, or is the cone of all real polynomials of degree not exceeding n that are nonnegative on a closed interval. Plentiful illustrative examples are provided. Some open problems are posed at the end.
Let K be a proper cone in R n , let A be an n×n real matrix that satisfies AK ⊆ K, let b be a giv... more Let K be a proper cone in R n , let A be an n×n real matrix that satisfies AK ⊆ K, let b be a given vector of K, and let λ be a given positive real number. The following two linear equations are considered in this paper: (i) (λI n − A)x = b, x ∈ K, and (ii) (A − λI n)x = b, x ∈ K. We obtain several equivalent conditions for the solvability of the first equation. For the second equation we give an equivalent condition for its solvability in case when λ > ρ b (A), and we also find a necessary condition when λ = ρ b (A) and also when λ < ρ b (A), sufficiently close to ρ b (A), where ρ b (A) denotes the local spectral radius of A at b. With λ fixed, we also consider the questions of when the set (A − λI n)K K equals {0} or K, and what the face of K generated by the set is. Then we derive some new results about local spectral radii and Collatz-Wielandt sets (or numbers) associated with a cone-preserving map, and extend a known characterization of M-matrices among Z-matrices in terms of alternating sequences.
Let K,, KS he cones. We say that K, is a subcone of Kz if EktK, cJW Kp Furthermore, if K1 # K2, K... more Let K,, KS he cones. We say that K, is a subcone of Kz if EktK, cJW Kp Furthermore, if K1 # K2, K1 is called a proper subcone; if dimK, =dim K2, K, is called a non-degenerate s&cone. We first prove that every n-dimensional indecomposable cone, n > 3, contains a non-degenerate indecomposable subcone which has no more than Zn-2 extremals. Then we construct for each n > 3 an n-dimensional indecomposable cone with exactly 2n-2 extremals such that each of its proper non-degenerate subcones is decomposable. *The contents of this paper formed part of the author's Ph.D. thesis at the University of Hong Kong, 1977. Thanks are due to Dr. Y. H. Au-Yeung for his advice and encouragement. 'After he had submitted the paper for publication, the author learned that a similar notion was also considered by Fiedler and P&k (see [3]).
J.H. Drew et al. [Linear and Multilinear Algebra 37 (1994) 304] conjectured that for n 4, the com... more J.H. Drew et al. [Linear and Multilinear Algebra 37 (1994) 304] conjectured that for n 4, the completely positive (CP) rank of every n × n completely positive matrix is at most [n 2 /4]. In this paper we prove that the CP rank of a 5 × 5 completely positive matrix which has at least one zero entry is at most 6, thus providing new supporting evidence for the conjecture.
We offer an almost self-contained development of Perron-Frobenius type results for the numerical ... more We offer an almost self-contained development of Perron-Frobenius type results for the numerical range of an (irreducible) nonnegative matrix, rederiving and completing the previous work of Issos, Nylen and Tam, and Tam and Yang on this topic. We solve the open problem of characterizing nonnegative matrices whose numerical ranges are regular convex polygons with center at the origin. Some related results are obtained and some open problems are also posed.
For every integer n≥4, it is proved that there is a unique graph of order n which maximizes the s... more For every integer n≥4, it is proved that there is a unique graph of order n which maximizes the spectral radius of the unoriented Laplacian matrix over all bicyclic graphs of order n, namely, the graph obtained from the cycle C4 by first adding a chord and then attaching n − 4 pendant edges to one end of the chord.
The purpose of this paper is to characterize those linear transformations on the space of $n \tim... more The purpose of this paper is to characterize those linear transformations on the space of $n \times n$ real matrices which map the class of $n \times n$ inverse $M$- matrices (or, the closure of this class) onto itself. As a by-product of our approach, we also obtain a sufficient condition for an inverse $M$-matrix (resp. $M$-matrix) to have all positive powers being inverse $M$-matrices (resp. $M$-matrices).
Cone-preserving map K-primitive matrix Exponents Polyhedral cone Exp-maximal cone Exp-maximal K-p... more Cone-preserving map K-primitive matrix Exponents Polyhedral cone Exp-maximal cone Exp-maximal K-primitive matrix Cone-equivalence Minimal cone Let K be a proper (i.e., closed, pointed, full convex) cone in R n. An n × n matrix A is said to be K-primitive if there exists a positive integer k such that A k (K \ {0}) ⊆ int K; the least such k is referred to as the exponent of A and is denoted by γ (A). For a polyhedral cone K, the maximum value of γ (A), taken over all K-primitive matrices A, is called the exponent of K and is denoted by γ (K). It is proved that the maximum value of γ (K) as K runs through all ndimensional minimal cones (i.e., cones having n + 1 extreme rays) is n 2 − n + 1 if n is odd, and is n 2 − n if n is even, the maximum value of the exponent being attained by a minimal cone with a balanced relation for its extreme vectors. The K-primitive matrices A such that γ (A) attain the maximum value are identified up to cone-equivalence modulo positive scalar multiplication.
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