Some inequalities for the Hadamard product and the Fan product of matrices

If A and B are nonsingular M-matrices, a sharp lower bound on the smallest eigenvalue τ(A★B) for the Fan product of A and B is given, and a sharp lower bound on τ(A°B^-1) for the Hadamard product of A and B^-1 is derived. In addition, we also give a sharp upper bound on the spectral radius ρ(A°B) for nonnegative matrices A and B.     

Schur product of matrices and numerical radius (range) preserving maps

Let F(A) be the numerical range or the numerical radius of a square matrix A. Denote by A ° B the Schur product of two matrices A and B. Characterizations are given for mappings on square matrices satisfying F(A ° B) = F(?(A) ° ?(B)) for all matrices A and B. Analogous results are obtained for mappings on Hermitian matrices.     

Preservers of eigenvalue inclusion sets

For a square matrix A, let S(A) be an eigenvalue inclusion set such as the Gershgorin region, the Brauer region in terms of Cassini ovals, and the Ostrowski region. Characterization is obtained for maps Φ on n×n matrices satisfying S(Φ(A)-Φ(B))=S(A-B) for all matrices A and B. From these results, one can deduce the structure of additive or (real) linear maps satisfying S(A)=S(Φ(A)) for every matrix A.     

On the spectral radius and the spectral norm of Hadamard products of nonnegative matrices

We prove the spectral radius inequality ρ(A₁°A₂°?°A_k)?ρ(A₁A₂?A_k) for nonnegative matrices using the ideas of Horn and Zhang. We obtain the inequality ‖A°B‖?ρ(A^TB) for nonnegative matrices, which improves Schur’s classical inequality ‖A°B‖?‖A‖‖B‖, where ‖·‖ denotes the spectral norm. We also give counterexamples to two conjectures about the Hadamard product.     

Preservers of eigenvalue inclusion sets of matrix products

For a square matrix A, let S(A) be an eigenvalue inclusion set such as the Gershgorin region, the union of Cassini ovals, and the Ostrowski’s set. Characterization is obtained for maps Φ on n×n matrices satisfying S(Φ(A)Φ(B))=S(AB) for all matrices A and B.     

Shorted operators of partitioned matrices and application

In this paper, our main objective is to study the effect of appending/deleting a column/row on the shorted operators. It turns out that for matrices A and B for which the shorted operator S(A|B) exists, S(A₁|B₁) of the matrix A₁=[A:a] with respect to the matrix B₁=[B:b], when it exists, is obtained by appending a suitable column to S(A|B). Moreover, if S(A₁|B₁) exists, then S(A|B) exists and is obtained from S(A₁|B₁) by dropping its last column. In the process, we study the effect of appending/deleting a column/row on the space pre-order and the parallel sum of parallel summable matrices. Finally, we specialize to the case of and matrices and study the effect of bordering (by an additional column and a row) on the shorted operator. We conclude the paper with an application to Linear Models with singular dispersion structure.     

Congruence of Hermitian matrices by Hermitian matrices

Two Hermitian matrices A,B∈M_n(C) are said to be Hermitian-congruent if there exists a nonsingular Hermitian matrix C∈M_n(C) such that B=CAC. In this paper, we give necessary and sufficient conditions for two nonsingular simultaneously unitarily diagonalizable Hermitian matrices A and B to be Hermitian-congruent. Moreover, when A and B are Hermitian-congruent, we describe the possible inertias of the Hermitian matrices C that carry the congruence. We also give necessary and sufficient conditions for any 2-by-2 nonsingular Hermitian matrices to be Hermitian-congruent. In both of the studied cases, we show that if A and B are real and Hermitian-congruent, then they are congruent by a real symmetric matrix. Finally we note that if A and B are 2-by-2 nonsingular real symmetric matrices having the same sign pattern, then there is always a real symmetric matrix C satisfying B=CAC. Moreover, if both matrices are positive, then C can be picked with arbitrary inertia.     

Bijective linear maps on semimodules spanned by Boolean matrices of fixed rank

Let M_m,n(B) be the semimodule of all m×n Boolean matrices where B is the Boolean algebra with two elements. Let k be a positive integer such that 2?k?min(m,n). Let B(m,n,k) denote the subsemimodule of M_m,n(B) spanned by the set of all rank k matrices. We show that if T is a bijective linear mapping on B(m,n,k), then there exist permutation matrices P and Q such that T(A)=PAQ for all A∈B(m,n,k) or m=n and T(A)=PA^tQ for all A∈B(m,n,k). This result follows from a more general theorem we prove concerning the structure of linear mappings on B(m,n,k) that preserve both the weight of each matrix and rank one matrices of weight k². Here the weight of a Boolean matrix is the number of its nonzero entries.     

Sign patterns that require almost unique rank

A sign pattern matrix is a matrix whose entries are from the set {+,-,0}. For a real matrix B, sgn(B) is the sign pattern matrix obtained by replacing each positive (respectively, negative, zero) entry of B by + (respectively, −, 0). For a sign pattern matrix A, the sign pattern class of A, denoted Q(A), is defined as {B:sgn(B)=A}. The minimum rank mr(A) (maximum rank MR(A)) of a sign pattern matrix A is the minimum (maximum) of the ranks of the real matrices in Q(A). Several results concerning sign patterns A that require almost unique rank, that is to say, the sign patterns A such that MR(A) = mr(A) + 1, are established and are extended to sign patterns A for which the spread is d=MR(A)-mr(A). A complete characterization of the sign patterns that require almost unique rank is obtained.     

Spectral radius of Hadamard product versus conventional product for non-negative matrices

We prove an inequality for the spectral radius of products of non-negative matrices conjectured by X. Zhan. We show that for all n×n non-negative matrices A and B, ρ(A°B)?ρ((A°A)(B°B))^1/2?ρ(AB), in which ° represents the Hadamard product.     

Preservers of pairs of bivectors with bounded distance

We extend Liu’s fundamental theorem of the geometry of alternate matrices to the second exterior power of an infinite dimensional vector space and also use her theorem to characterize surjective mappings T from the vector space V of all n×n alternate matrices over a field with at least three elements onto itself such that for any pair A, B in V, rank(A-B)?2k if and only if rank(T(A)-T(B))?2k, where k is a fixed positive integer such that n?2k+2 and k?2.     

Bounds on eigenvalues of the Hadamard product and the Fan product of matrices

We prove an upper bound for the spectral radius of the Hadamard product of nonnegative matrices and a lower bound for the minimum eigenvalue of the Fan product of M-matrices. These improve two existing results.     

Imbedding conditions for normal matrices

When can an (n-k)×(n-k) normal matrix B be imbedded in an n×n normal matrix A? This question was studied for the first time 50 years ago by Ky Fan and Gordon Pall, who gave the complete answer in the case k=1. Since then, a few authors obtained additional results. In this note, we show how an approach inspired by the Hermitian case can throw some light on the problem.     

On extremal matrices of second largest exponent by Boolean rank

Let b = b(A) be the Boolean rank of an n × n primitive Boolean matrix A and exp(A) be the exponent of A. Then exp(A) ? (b − 1)² + 2, and the matrices for which equality occurs have been determined in [D.A. Gregory, S.J. Kirkland, N.J. Pullman, A bound on the exponent of a primitive matrix using Boolean rank, Linear Algebra Appl. 217 (1995) 101-116]. In this paper, we show that for each 3 ? b ? n − 1, there are n × n primitive Boolean matrices A with b(A) = b such that exp(A) = (b − 1)² + 1, and we explicitly describe all such matrices.     

On D. Hägele’s approach to the Bessis-Moussa-Villani conjecture

The reformulation of the Bessis-Moussa-Villani (BMV) conjecture given by Lieb and Seiringer asserts that the coefficient α_m,k(A,B) of t^k in the polynomial Tr(A+tB)^m, with A,B positive semidefinite matrices, is nonnegative for all m,k. We propose a natural extension of a method of attack on this problem due to Hägele, and investigate for what values of m,k the method is successful, obtaining a complete determination when either m or k is odd.     

On the span of Hadamard products of vectors

Let A₁, … , A_k be positive semidefinite matrices and B₁, … , B_k arbitrary complex matrices of order n. We show that

span{(A₁x)°(A₂x)°?°(A_kx)|x∈Cⁿ}=range(A₁°A₂°?°A_k)     

Pairs of mutually annihilating operators

Pairs (A,B) of mutually annihilating operators AB=BA=0 on a finite dimensional vector space over an algebraically closed field were classified by Gelfand and Ponomarev [Russian Math. Surveys 23 (1968) 1-58] by method of linear relations. The classification of (A,B) over any field was derived by Nazarova, Roiter, Sergeichuk, and Bondarenko [J. Soviet Math. 3 (1975) 636-654] from the classification of finitely generated modules over a dyad of two local Dedekind rings. We give canonical matrices of (A,B) over any field in an explicit form and our proof is constructive: the matrices of (A,B) are sequentially reduced to their canonical form by similarity transformations (A,B)?(S^-1AS,S^-1BS).     

On classical adjoint-commuting mappings between matrix algebras

Let F be a field and let m and n be integers with m,n?3. Let M_n denote the algebra of n×n matrices over F. In this note, we characterize mappings ψ:M_n→M_m that satisfy one of the following conditions:

1.

|F|=2 or |F|>n+1, and ψ(adj(A+αB))=adj(ψ(A)+αψ(B)) for all A,B∈M_n and α∈F with ψ(I_n)≠0.

2.

ψ is surjective and ψ(adj(A-B))=adj(ψ(A)-ψ(B)) for every A,B∈M_n.

Here, adjA denotes the classical adjoint of the matrix A, and I_n is the identity matrix of order n. We give examples showing the indispensability of the assumption ψ(I_n)≠0 in our results.     

To what extent is a large space of matrices not closed under product?

Let (K) be a field. Given an arbitrary linear subspace V of M_n(K) of codimension less than n-1, a classical result states that V generates the (K)-algebra M_n(K). Here, we strengthen this statement in three ways: we show that M_n(K) is spanned by the products of the form AB with (A,B)∈V²; we prove that every matrix in M_n(K) can be decomposed into a product of matrices of V; finally, when V is a linear perplane of M_n(K) and n>2, we show that every matrix in M_n(K) is a product of two elements of V.     

A note on “Constructing matrices with prescribed off-diagonal submatrix and invariant polynomials”

Let F be any field and let B a matrix of F^q×p. Zaballa found necessary and sufficient conditions for the existence of a matrix A=[A_ij]_i,j∈{1,2}∈F^(p+q)×(p+q) with prescribed similarity class and such that A₂₁=B. In an earlier paper [A. Borobia, R. Canogar, Constructing matrices with prescribed off-diagonal submatrix and invariant polynomials, Linear Algebra Appl. 424 (2007) 615-633] we obtained, for fields of characteristic different from 2, a finite step algorithm to construct A when it exists. In this short note we extend the algorithm to any field.