Some research problems on the hadamard product and singular values of matrices

We pose some problems on the Hadamard product and singular values of matrices.     

Some research problems on the hadamard product and singular values of matrices

We pose some problems on the Hadamard product and singular values of matrices.     

Spectrally arbitrary zero-nonzero patterns of order 4

In this article the authors characterize all the 4×4 zero-nonzero patterns that are spectrally arbitrary. Several observations and conjectures are presented for the n × n case.     

Some necessary and some sufficient trace inequalities for Euclidean distance matrices

In this article, we use known bounds on the smallest eigenvalue of a symmetric matrix and Schoenberg's theorem to provide both necessary as well as sufficient trace inequalities that guarantee a matrix D is a Euclidean distance matrix, EDM . We also provide necessary and sufficient trace inequalities that guarantee a matrix D is an EDM generated by a regular figure.     

Some necessary and some sufficient trace inequalities for Euclidean distance matrices

In this article, we use known bounds on the smallest eigenvalue of a symmetric matrix and Schoenberg's theorem to provide both necessary as well as sufficient trace inequalities that guarantee a matrix D is a Euclidean distance matrix, EDM. We also provide necessary and sufficient trace inequalities that guarantee a matrix D is an EDM generated by a regular figure.     

Construction of nonnegative matrices and the inverse eigenvalue problem

This article presents a technique for combining two matrices, an n?×?n matrix M and an m?×?m matrix B, with known spectra to create an (n?+?m???p)?×?(n?+?m???p) matrix N whose spectrum consists of the spectrum of the matrix M and m???p eigenvalues of the matrix B. Conditions are given when the matrix N obtained in this construction is nonnegative. Finally, these observations are used to obtain several results on how to construct a realizable list of n?+?1 complex numbers (λ₁,λ₂,λ₃,σ) from a given realizable list of n complex numbers (c ₁,c ₂,σ), where c ₁ is the Perron eigenvalue, c ₂ is a real number and σ is a list of n???2 complex numbers.     

Signature preserving linear maps of Hermitian matrices

The structure of a unital linear map on hermitian matrices with the property that it preserves the set of invertible hermitian matrices with fixed indefinite inertia is examined. It turns out that such a map is either a unitary similarity or a unitary similarity followed by a transposition (the case when the fixed inertia has equal number of positive and negative eigenvalues is excluded).     

Matrices that preserve vectors of fixed sign variation

For each k≥ 0, those nonsingular matrices that transform the set of totally nonzero vectors with k sign variations into (respectively, onto) itself are studied. Necessary and sufficient conditions are provided. The cases k=0,1,2,n-3,n-2,n-1 are completely characterized.     

Some rank equalities and inequalities for Kronecker products of matrices

A set of rank equalities and inequalities are established for block matrices consisting of Kronecker products. Various consequences are also given.     

Some rank equalities and inequalities for Kronecker products of matrices

A set of rank equalities and inequalities are established for block matrices consisting of Kronecker products. Various consequences are also given.     

Eigenvalues of products of matrices

Let A and B be n?×?n matrices over an algebraically closed field F. The pair ( A,?B ) is said to be spectrally complete if, for every sequence c₁,…,c_n ∈F such that det (AB)=c₁ ,…,c_n , there exist matrices A′,B,′∈F,^n×n similar to A,?B, respectively, such that A′B′ has eigenvalues c₁,…,c_n . In this article, we describe the spectrally complete pairs. Assuming that A and B are nonsingular, the possible eigenvalues of A′B′ when A′ and B′ run over the sets of the matrices similar to A and B, respectively, were described in a previous article.     

Bounds for the zeros of polynomials from matrix inequalities - II

In this article, we present new bounds for the zeros of polynomials depending on some estimates for the spectral norms and the spectral radii of the square and the cube of the Frobenius companion matrix.     

Rank-one nonincreasing mappings on triangular matrices and some related preserver problems

Let T_n (F) be the algebra of all n×n upper triangular matrices over an arbitrary field F. We first characterize those rank-one nonincreasing mappings ψ: T_n (F)→T_m (F)n?m such that ψ(I_n ) is of rank n. We next deduce from this result certain types of singular rank-one r-potent preservers and nonzero r-potent preservers on T_n (F). Characterizations of certain classes of homomorphisms and semi-homomorphisms on T_n (F) are also given.     

The generalized Schur complement in group inverses and (k 1)-potent matrices

In this article, two facts related to the generalized Schur complement are studied. The first one is to find necessary and sufficient conditions to characterize when the group inverse of a partitioned matrix can be expressed in the Schur form. The other one is to develop a formula for any power of the generalized Schur complement of an idempotent partitioned matrix and then to characterize when this generalized Schur complement is a (k+1)-potent matrix. In addition, some spectral theory related to this complement is analyzed.     

Dimension inequalities for the closure of the image of a multilinear function

In this note we give a simple proof and an extension of a dimension inequality of Howard concerning the range of a multilinear function with vector space range by using some results on algebraic varieties.     

Dimension inequalities for the closure of the image of a multilinear function

In this note we give a simple proof and an extension of a dimension inequality of Howard concerning the range of a multilinear function with vector space range by using some results on algebraic varieties.     

A result about determinantal divisors

Let Rbe a principal ideal ringR_n the ring of n× nmatrices over R, and d_k (A) the kth determinantal divisor of Afor 1 ? k? n, where Ais any element of R_n , It is shown that if A,BεR_n , det(A) det(B:) ≠ 0, then d_k (AB) ≡ 0 mod d_k (A) d_k (B). If in addition (det(A), det(B)) = 1, then it is also shown that d_k (AB) = d_k (A) d_k (B). This provides a new proof of the multiplicativity of the Smith normal form for matrices with relatively prime determinants.