About
103
Publications
4,048
Reads
How we measure 'reads'
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more
1,810
Citations
Publications
Publications (103)
This extended Oberwolfach report (to appear in the proceedings of the MFO Workshop 2335: Aspects of Aperiodic Order) announces the full solution to the Dry Ten Martini Problem for Sturmian Hamiltonians. Specifically, we show that all spectral gaps of Sturmian Hamiltonians (as predicted by the gap labeling theorem) are open for all nonzero couplings...
We consider polynomials in R[x] which map the set of nonnegative (element-wise) matrices of a given order into itself. Let n be a positive integer and define P(n)= {p in R[x] : p(A) is nonnegative (element-wise), for all A, A an n-by-n nonnegative (element-wise) matrix}. This set plays a role in the Nonnegative Inverse Eigenvalue Problem. Clark and...
Given an n×n real symmetric matrix A, let λn(A) denote its smallest eigenvalue. Let Kn denote the set of all n×n invertible, lower triangular (0,1) matrices, and cn:=min{λn(YYt):Y∈Kn}. Then, cn is the smallest singular value in Kn. Hong and Loewy introduced cn as a mean to obtain inequalities involving eigenvalues of certain GCD (greatest common di...
Let A be a nonnegative symmetric 5×5 matrix with eigenvalues λ1≥λ2≥λ3≥λ4≥λ5. Let s1=∑i=15λi and s3=∑i=15λi3. One of the well-known JLL inequalities states that 25s3≥s13. We show that if λ3≥s1 this JLL inequality can be strengthened to s3≥s13. Further, we show that if λ2=λ3, λ4=λ5 and λ5<72λ2−52λ1 then λ5≥−12λ1+λ2. This allows us to show that certai...
The Real Nonnegative Inverse Eigenvalue Problem (RNIEP) asks when is a list \[ \sigma=(\lambda_1, \lambda_2,\ldots,\lambda_n)\] consisting of real numbers the spectrum of an $n \times n$ nonnegative matrix $A$. In that case, $\sigma$ is said to be realizable and $A$ is a realizing matrix. In a recent paper dealing with RNIEP, P.~Paparella considere...
A linear map $L$ from ${\mathbb C}^{n \times n}$ into ${\mathbb C}^{n \times n}$ is called a quantum channel if it is completely positive and trace preserving. The set ${\cal L}_n$ of all such quantum channels is known to be a compact convex set. While the extreme points of ${\cal L}_n$ can be characterized, not much is known about the structure of...
Let $A$ be a nonnegative symmetric $ 5 \times 5 $ matrix with eigenvalues $ \lambda_1 \geq \lambda_2 \geq \lambda_3 \geq \lambda_4 \geq \lambda_5 $. We show that if $ \sum_{i=1}^{5} \lambda_{i} \geq \frac{1}{2} \lambda_1 $ then $ \lambda_3 \leq \sum_{i=1}^{5} \lambda_{i} $. McDonald and Neumann showed that $ \lambda_1 + \lambda_3 + \lambda_4 \geq 0...
Let $A$ be a nonnegative symmetric $ 5 \times 5 $ matrix with eigenvalues $ \lambda_1 \geq \lambda_2 \geq \lambda_3 \geq \lambda_4 \geq \lambda_5 $. We show that if $ \sum_{i=1}^{5} \lambda_{i} \geq \frac{1}{2} \lambda_1 $ then $ \lambda_3 \leq \sum_{i=1}^{5} \lambda_{i} $. McDonald and Neumann showed that $ \lambda_1 + \lambda_3 + \lambda_4 \geq 0...
Let L(m,n) denote the convex set of completely positive trace preserving
operators from m by m to n by n complex valued matrices, i.e quantum channels.
We give a necessary condition for L in L(m,n) to be an extreme point. We show
that generically, this condition is also sufficient.
We characterize completely the extreme points of L(2,2), i.e. quant...
This article is dedicated to the memory of Professor Uriel George Rothblum who passed away unexpectedly on March 26, 2012. Professor Rothblum was known to all of us as Uri. Following some biographical details, a survey of some of Uri’s research contributions is given. In light of Uri’s extensive list of publications and large number of coauthors th...
Let A be an nxn (entrywise) positive matrix and let f(t)=det(I-t A). We prove
that there always exists a positive integer N such that 1-f(t)^{1/N} has
positive coefficients.
Let A be an n×n irreducible nonnegative (elementwise) matrix. Borobia and Moro raised the following question: Suppose that every diagonal of A contains a positive entry. Is A similar to a positive matrix? We give an affirmative answer in the case n = 4.
In this paper we deal with two aspects of the minimum rank of a simple undirected graph $G$ on $n$ vertices over a finite field $\FF_q$ with $q$ elements, which is denoted by $\mr(\FF_q,G)$. In the first part of this paper we show that the average minimum rank of simple undirected labeled graphs on $n$ vertices over $\FF_2$ is $(1-\varepsilon_n)n$,...
Let K be a proper (i.e., closed, pointed, full convex) cone in Rn. An n×n matrix A is said to be K-primitive if there exists a positive integer k such that Ak(K∖{0})⊆intK; 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...
Let G be an undirected graph on n vertices and let S(G) be the set of all real symmetric n×n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The inverse inertia problem for G asks which inertias can be attained by a matrix in S(G). We give a complete answer to this question for trees in te...
We discuss the converse of a theorem of Potter stating that if the matrix equation AB=ωBA is satisfied with ω a primitive qth root of unity, then Aq+Bq=(A+B)q. We show that both conditions have to be modified to get a converse statement and we present a characterization when the converse holds for these modified conditions and q=3 and a conjecture...
Let $K$ be a proper (i.e., closed, pointed, full convex) cone in ${\Bbb R}^n$. An $n\times n$ matrix $A$ is said to be $K$-primitive if there exists a positive integer $k$ such that $A^k(K \setminus \{0 \}) \subseteq$ int $K$; the least such $k$ is referred to as the exponent of $A$ and is denoted by $\gamma(A)$. For a polyhedral cone $K$, the maxi...
We study the question in which nonnegative mat rices are similar to positive matrices first raised by Borobia and Moro in 1997. We identify certain classes of nonnegative matrices that are not similar to positive matrices and solve the problem when an n X n nonnegative matrix has exactly n or n +1 elements equal to zero.
Given an arbitrary strictly increasing infinite sequence {x i} ∞i=1 of positive integers, let S n {x 1,⋯, x n} for any integer n ≥ 1. Let q ≥ 1 be a given integer and f an arithmetical function. Let λ (1)n ≤ ⋯ ≤ λ (n)n be the eigenvalues of the matrix (f(x i, x j)) having f evaluated at the greatest common divisor (x i, x j) of x i and x j as its i...
The symmetric nonnegative inverse eigenvalue problem (SNIEP) asks when a list σ = (λ 1 , λ 2 , . . . , λ n) of n real numbers is the spectrum of an n×n symmetric nonnegative matrix. This problem is completely solved only for n ≤ 4. Our main goal here is to contribute to the solution of SNIEP for n = 5. We also give a sufficient condition for a list...
Let G be an undirected graph on n vertices and let S(G) be the set of
all real symmetric n x n matrices whose nonzero off-diagonal entries
occur in exactly the positions corresponding to the edges of G. The
inverse inertia problem for G asks which inertias can be attained by a
matrix in S(G). We give a complete answer to this question for trees in...
Characterized are all simple undirected graphs $G$ such that any real symmetric matrix that has graph $G$ has no eigenvalues of multiplicity more than 2. All such graphs are partial 2-trees (and this follows from a result for rather general fields), but only certain partial 2-trees guarantee maximum multiplicity 2. Among partial linear 2-trees, the...
Our main result is a sharp bound for the number of vertices in a minimal forbidden subgraph for the graphs having minimum rank at most 3 over the finite field of order 2. We also list all 62 such minimal forbidden subgraphs. We conclude by exploring how some of these results over the finite field of order 2 extend to arbitrary fields and demonstrat...
Let F be a finite field, G =( V, E) be an undirected graph on n vertices, and let S(F, G) be the set of all symmetric n × n matrices over F whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G.L et mr(F, G) be the minimum rank of all matrices in S(F, G). If F is a finite field with pt elements, p � =2 ,...
Let F be a field, G =( V, E) be an undirected graph on n vertices, and let S(F, G) be the set of all symmetric n × n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. For example, if G is a path, S(F, G )co nsists of the symmetric irreducible tridiagonal matrices. Let mr(F, G) be the minimum...
Let $\{x_i\}_{i=1}^{\infty}$ be an arbitrary strictly increasing infinite sequence of positive integers. For an integer $n\ge 1$, let $S_n=\{x_1,\ldots,x_n\}$. Let $\varepsilon$ be a real number and $q\ge 1$ a given integer. Let \smash{$\lambda _n^{(1)}\le \cdots\le \lambda _n^{(n)}$} be the eigenvalues of the power GCD matrix $((x_i, x_j)^{\vareps...
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 [n2/4]. 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...
In (1) the recursive inverse eigenvalue problem for matrices was introduced. In this paper we examine an open problem on the existence of symmetric positive semidefinite solutions that was posed there. We first give several counterexamples for the general case and then characterize under which further assumptions the conjecture is valid.
Let Mn(R) and Sn(R) be the spaces of n × n real matri- ces and real symmetric matrices respectively. We continue to study d(n, n − 2, R): the minimal numbersuch that every � -dimensional subspace of Sn(R) contains a nonzero matrix of rank n−2 or less. We show that d(4, 2, R) = 5 and obtain some upper bounds and monotonicity properties of d(n, n − 2...
Let k and n be positive integers such that k⩽n. Let Sn(F) denote the space of all n×n symmetric matrices over the field F with char F≠2. A subspace L of Sn(F) is said to be a k-subspace if rank A⩽k for every AεL. Now suppose that k is even, and write k=2r. We say a k∥-subspace of Sn(F) is decomposable if there exists in F a subspace W of dimension...
A function f from the symmetric group Sn into R is called a class function if it is constant on each conjugacy class. Let df be the generalized matrix function associated with f, mapping the n-by-n Hermitian matrices to R. For example, if f(σ)=sgn(σ), then df(A)=detA. Let Kn(Kn(R)) denote the closed convex cone of those f for which df(A)⩾0 for all...
A function f from the symmetric group Sn into R is called a class function if f(σ) = f(τ) whenever τ is conjugate to σ. Let df be thegeneralized matrix function associated with f, mapping the n-by-n positive semidefinite Hermitian matricesto R. For example, if f(σ) = sgn(σ), then df(A) = det A.
We consider the cone Kn of those f for which df(A) ≥ 0...
Let Sn(F) denote the space of all n × n symmetric matrices over the field F. Given a positive integer k such that k < n, let d(n, k, F) be the smallest integer ℓ such that every ℓ dimensional subspace of Sn(F) contains a nonzero matrix whose rank is at most k. It is our purpose to consider d(n, k, F) for ℝ and ℂ. While the computation of d(n, k, Fℂ...
Among various matrix completion problems that have been considered in recent years, the positive definite completion problem seems to have received the most attention. Indeed, in addition to being a problem of great interest, it is related to various applications as well as other completion problems. It may also be viewed as a fundamental problem i...
The following problem, originally proposed by Omladič and Šemrl [Linear Algebra Appl., 249:29-46 (1996)], is considered. Let k and n be positive integers such that k < n. Let L be a subspace of M n(F), the space of n x n matrices over a field F, such that each A ε L has at most k distinct eigenvalues (in the algebraic closure of F). Then, what is t...
Let A be an nx complex matrix with inertia In(A)= (pi(A), theta(A), delta(A)), and let H be an n x n hermitian matrix with inertia In(H)= (pi(H), theta(H), delta(H)). Let K be an n x n positive semidefinite matrix such that K = AH + HA*. Suppose that l is the dimension of the controllability space of the pair (A, K). Lerer and Rodman conjectured th...
LetSn(F) denote the set of all n × n symmetric matrices over the field F. Let k be a positive integer such that k ≤ n. Alinear operator T on Sn(F) is said to be a rank-k preserver provided that it maps the set of all rank k matrices into itself. We show here that if k is even and F is algebraically closed of characteristic ≠ 2, then any such T must...
We show that there exist real numbers λ 1 ,λ 2 ,⋯,λ n that occur as the eigenvalues of an entry-wise nonnegative n-by-n matrix but do not occur as the eigenvalues of a symmetric nonnegative n-by-n-matrix. This solves a problem posed by M. Boyle and D. Handelman [Trans. Am. Math. Soc. 336, No. 1, 121-149 (1993; Zbl 0766.15024)], D. Hershkowitz [Exis...
Let A be an n-by-n irreducible, entrywise nonnegative matrix. For a given t > 0, we consider the problem of maximizing the Perron root of a nonnegative, diagonal, trace t perturbation of A. Because of the convexity of the Perron root as a function of diagonal entries, the maximum occurs for some tEii. Such an index i, which is called a winner, may...
Let V be one of the following four real vector spaces: n, the n × n real symmetric matrices; n, the n × n complex hermitian matrices; , the n × n real matrices, and , the n × n complex matrices. Suppose T is an -linear map on V preserving the invertible matrices in the case or or preserving the nonsingular balanced inertia class (n even) in the cas...
Let In denote the space of all n×n symmetric matrices over a field F. Let t be a positive integer such that t<n. A subspace W of In(F) is said to be a t̄-subspace provided that the rank of every matrix in W is bounded by t. Meshulam showed, under the assumption |F|≥n+ 1, that the maximal dimension of a t̄-subspace of In(F) is given by max t+1 2, k+...
We prove the following result. Let F be an infinite field of characteristic other than two. Let k be a positive integer. Let Sn(F) denote the space of all n × n symmetric matrices with entries in F, and let T: Sn(F) → Sn(F) be a linear operator. Suppose that T is rank-k nonincreasing and its image contains a matrix with rank higher than k. Then, th...
We prove the following result. Let F be an algebraically closed field and F be the space of all m×n matrices with entries in F. Let k be a positive integer. Let T : F → F be a linear transformation whose image contains a matrix with rank higher than k. Suppose also that if A ∈ F is any matrix of rank k then T(A) has rank less than or equal to k. Th...
Let M
n
(F) denote the set of n-by-n matrices over the field F. We consider the following question: Among matrices A ∈ M
n
(F) with rank A = k < n, how many diagonal entries of A must be changed, at worst, in order to guarantee that the rank of A is increased. Our initial motivation arose from an error pointed out in [BOvdD], but we also view this...
Let Vdenote either the space of n×n hermitian matrices or the space of n×nreal symmetric matrices, Given nonnegative integers r,s,t such that r+S+t=n, let G( r,s,r) denote the set of all matrices in V with inertia (r,s,t). We consider here linear operators on V which map G(r,s,t) into itself.
Maximal matrices within a given class of matrices are characterized for the star order, the minus order, and the Loewner partial order.
Let Kn be the convex set of n×npositive semidefinite doubly stochastic matrices. If A kn , the graph of A,G(A), is the graph on n vertices with (i,j) an edge if aij ≠ 0i≠ j. We are concerned with the extreme points in Kn . In many cases, the rank of Aand G(A) are enough to determine whether A is extreme in Kn . This is true, in particular, if G(A)i...
Let x and y be positive vectors in n. The set of all n × n nonnegative matrices having x and yT as their right and left Perron eigenvectors is a polyhedral convex cone. A cross section of this cone is the polytope consisting of all n × n nonnegative matrices C such that Cx=x and yTC=yT. The set of doubly stochastic matrices is obtained as a special...
Given AεMn(C) and BεMn,k(C) all possible inertias occurring in the Hermitian part of A+BX are determined as X runs over Mk,n(C).
Let n be an even integer such that n ⩾ 4. Let T be an invertible linear map on the space of n × n real symmetric matrices which maps the set of matrices having inertia( ,, 0) into itself. Then there exist a nonsingular matrix S and ϵ = ± 1 such that T(A) = ϵStAS. This is an analogue of a result obtained for Hermitian matrices by Pierce and Rodman.
Let V be a space of matrices which is one of the following three types. (I) V = Vm,n , the space of all m × n matrices with entries in a field F. We assume that F is an infinite field, and that m≤n. (II) V= , the space of all n × n complex hermitian matrices. (III) , the space of all n × n real symmetric matrices. Given a positive integer k and a l...
The purpose of this paper is to prove the following result: Let $n\geqq 3$ and let r, s be given positive integers such that $r \ne s$ and $r + s\leqq n$. Let $\mathcal{H}_n $ denote the space of all $n \times n$ hermitian matrices. Suppose that $T:\mathcal{H}_n \to \mathcal{H}_n $ is a linear transformation that maps the set of all matrices with r...
We describe some results concerning a linear transformation on a space V of matrices, which is rank preserving or rank nonincreasing on a certain subset of V.
This paper is divided into two parts. In the first part, suppose that K1 and K2 are proper cones and that A is a rank r linear transformation which maps the set of extremals of K1 into the set of extremals of K2. We give an upper bound for the dimension of the face generated by A in the cone Π(K1, K2). In the second part, we consider an indecomposa...
In a previous work we derived for the quark mass matrices in the standard model with any number of families, n, a set of basis-independent conditions, which are necessary and sufficient for CP invariance. Very recently Jarlskog claimed to have reduced the number of these conditions for n>3 to (1/2(n-1)(n-2) = the number of Kobayashi-Maskawa-type ph...
Let K1 and K2 be proper cones in the finite dimensional real vector spaces V1 and V2 respectively, and let ∏(K1,K2) be the cone of all linear transformations from V1 to V2 that map K1 into K2. For i = 1,2, let Ext Ki be the subset of Ki consisting of 0 and the extremals of Ki. Let Aϵ∏(K1,K2). Our purpose is to give an upper bound for the dimension...
It is our purpose to compute the maximum value of the modulus of the determinant of an m×m nonprincipal submatrix of an n×n hermitian (or real symmetric) matrix A, in terms of m, the eigenvalues of A, and the cardinality k of the set of common row and column indices of this submatrix.
Barker proved that the lattice of faces, (K), of a finite dimensional proper cone K is always complemented. The proof, however, contained a gap. In this paper we offer a correct proof of the result. At the same time new characterizations of lattices of finite length which are section complemented or relatively complemented are found. It is also pro...
Let A, B be n × n matrices with entries in a field F. Our purpose is to show the following theorem: Suppose n⩾4, A is irreducible, and for every partition of {1,2,…,n} into subsets α, β with ¦α¦⩾2, ¦β¦⩾2 either rank A[α¦β]⩾2 or rank A[β¦α]⩾2. If A and B have equal corresponding principal minors, of all orders, then B or Bt is diagonally similar to...
We derive a set of invariant quantities in fermion mass matrices, independent of one's weak-eigenstate basis, the vanishing of which is both necessary and sufficient for CP invariance. Our method is applied to the standard single-Higgs-doublet SU(2) x U(1) model with an arbitrary number of fermion generations.
Suppose q ≥ 2 and I is a field with at least q elements. Let Ar,= 1,….be nonzero linear maps (with appropriate vector spaces over F as domains and codomains), at most one of rank one. and let Br.i = 1…,q, be linear maps so that At and Bt have the same domain and codomain. Then each of (i)B1x⊗ ⋯ ⊗ Bqx = A1x ⊗ ⋯ ⊗ Aqxfor all x; (ii)B1x1v ⋯ v Bqxq = A...
Let F be a field and let {d 1,…,dk } be a set of independent indeterminates over F. Let A(d 1,…,dk ) be an n × n matrix each of whose entries is an element of F or a sum of an element of F and one of the indeterminates in {d 1,…,dk }. We assume that no d 1 appears twice in A(d 1,…,dk ). We show that if det A(d 1,…,dk ) = 0 then A(d 1,…,dk ) must co...
We compute here the maximum value of the modulus of the determinant of an m×m nonprincipal submatrix of an n×n positive semidefinite matrix A, in terms of m, the eigenvalues of A, and cardinality k of the set of common row and column indices of this submatrix.
Let A be an n × n normal matrix, and let 1 ⪕ m < n. Let α,β ϵ Qm,n, the set of increasing integer sequences of length m chosen from 1, 2,…, n. Suppose α and β have exactly k common entries, denoted by ∥α ∩ β∥ = k, and suppose k ⪕ m − 1. Marcus and Filippenko obtained an upper bound for ∥det A[α∥β]∥, which depends on k and the moduli of the eigenval...
It is known that, if T is an n × n complex matrix such that every characteristic root of UT has modulus I for every n × n unitary matrix U then T must be unitary. This paper generalizes this result in two directions, one of which provides a proof of a 1971 conjecture of M. Marcus. An analogous self-duaiity result is given for hermitian matrices, an...
Conditions are given on maps A, C∈ L(X, V) and B, D ∈ L(Y, V) for which Cx∧Dy = Ax∧By for all x ∈ X and y ∈ Y, and, when X = Y, for which Cx∧Dx = Ax ∧Bx for all x ∈ X.
A bilinear map on vectors induces several bilinear maps on linear maps. The extent to which certain uniqueness properties of the original map carry over to the induced maps is studied. Several results on matrix equations are obtained by specialization to matrix settings.
Let V be an n-dimensional Euclidean vector space, and let V(m) be the corresponding m-th completely symmetric space over V equipped with the induced inner product. The purpose of this paper is to prove the following conjecture of H.A. Robinson: if T is a linear operator on V(m) and (Tz, z) = 0 for every decomposable element z of V(m), then T is ske...
Given an arbitrary n×n matrix A with complex entries, we characterize all inertia triples (abc) that are attained by the Lyapunov transform AH+ HA , as H varies over the set of all n× n positive definite matrices.
Let be the space of n×n real symmetric matrices and let +n be the cone of n×n positive semidefinite matrices in n. If A∈Rn,n, then the real Lyapunov transformation A : n→n corresponding to A is defined by . We prove that given A,B∈Rn,n such that A is invertible, we have if and only if B=μA or B=μA−1 for some μ > 0, and that this condition is also n...
Let ℂ n,n denote the space of n × n matrices with complex entries and let ℋ n denote the set of n × n hermitian matrices. Given any matrix A ∊ℂ n,n , the Lyapunov transformation corresponding to A is defined by ℐ A ( H ) = AH + HA * , where H ∊ℋ n . Let PSD ( n ) be the set of all n × n hermitian positive semidefinite matrices. Taussky [8, 9] raise...
Let u be an (r - l)(2n - r + 2)/2 dimensional subspace of n × n real valued symmetric matrices. Then u contains a nonzero matrix whose greatest eigenvalue is at least of multiplicity r, if 2≦ r ≦ n – 1. This bound is best possible. We apply this result to prove the Bohnenblust generalization of Calabi’s theorem. We extend these results to hermitian...
We study some relations between a reproducing cone K in a linear space V over a fully ordered field F and the cone Γ(K) in Hom (V,V) consisting of all operators A such that AK ⊆ K. In particular, indecomposable cones are considered.
If K is a cone in Rn we let Γ(K) denote the cone in the space Mn of nXn matrices consisting of all A such that AK ⊆ K. We show first that Γ(K) is indecomposable if and only if K is indecomposable. Next we let Γ(K)∗ be the dual of Γ(K). Then we show that Γ(K)=Γ(K)∗ if and only if K is the image of the nonnegative orthant under an orthogonal transfor...
Let be the n-dimensional ice cream cone, and let Γ(Kn) be the cone of all matrices in nn mapping Kn into itself. We determine the structure of Γ(Kn), and in particular characterize the extreme matrices in Γ(Kn).
The Lyapunov transformation corresponding to the matrix is a linear transformation on the space of hermitian matrices of the form Given a positive stable , the Stein-Pfeffer Theorem characterizes those where B is similar to A and H is positive definite. Here several extensions of this theorem are proved
A refined Love-type theory of motion is established for orthotropic composite cylindrical shells. An extensional-rotational dynamic coupling effect is shown to exist, expressed by R1 inertia terms. An extendedversion of the theoryis formulated to account for dynamic stability problems involving time-dependent and non-conservative forcesThe frequenc...