Invariant factors,Wiener–Hopf factorization indices,and invariant factors at infinity of rational matrices

The relationship between the finite structure, the infinite structure, and the Wiener–Hopf factorization indices of any rectangular rational matrix is studied.     

Rank reduction, factorization and conjugation

We investigate the rank reduction procedure, related factorizations and conjugation algorithms. An exact characterization of the full rank factorization produced by the rank reduction algorithm is given. This result is then used to derive matrix decompositions and conjugation procedures.     

Inequalities for hadamard product and unitarily invariant norms of matrices

The paper contains some general theorems for Hadamard product of matrices which in particular include Fiedler's Theorem and a better bound for an inequality on product of eigenvalues of certain matrices due to Ando. Lieb's concavity Theorem has been proved using operator means. Some inequalities for unitarily invariant norms have also been proved.     

Inequalities for hadamard product and unitarily invariant norms of matrices

The paper contains some general theorems for Hadamard product of matrices which in particular include Fiedler's Theorem and a better bound for an inequality on product of eigenvalues of certain matrices due to Ando. Lieb's concavity Theorem has been proved using operator means. Some inequalities for unitarily invariant norms have also been proved.     

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.     

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.     

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.     

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.     

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.     

On the generalized indices of boolean matrices

We characterize completely those Boolean matrices with the largest generalized indices in the class of Boolean matrices and in the class of reducible Boolean matrices and derive a new upper bound for the generalized index in terms of period. We also generalize the upper and lower multiexponents of primitive Boolean matrices to general Boolean matrices.     

On the generalized indices of boolean matrices

We characterize completely those Boolean matrices with the largest generalized indices in the class of Boolean matrices and in the class of reducible Boolean matrices and derive a new upper bound for the generalized index in terms of period. We also generalize the upper and lower multiexponents of primitive Boolean matrices to general Boolean matrices.     

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.     

The Moore-Penrose inverse for sums of matrices under rank additivity conditions

Some results on the Moore-Penrose inverse for sums of matrices under rank additivity conditions are revisited and some new consequences are presented. Their extensions to the weighted Moore-Penrose inverse of sums of matrices under rank additivity conditions are also considered.     

Singularities of rational functions and minimal factorizations: The noncommutative and the commutative setting

We show that the singularities of a matrix-valued noncommutative rational function which is regular at zero coincide with the singularities of the resolvent in its minimal state space realization. The proof uses a new notion of noncommutative backward shifts. As an application, we establish the commutative counterpart of the singularities theorem: the singularities of a matrix-valued commutative rational function which is regular at zero coincide with the singularities of the resolvent in any of its Fornasini-Marchesini realizations with the minimal possible state space dimension. The singularities results imply the absence of zero-pole cancellations in a minimal factorization, both in the noncommutative and in the commutative setting.     

Positive extension and completion problems for a class of structured matrices

A class of matrices, defined by a displacement rank property, is introduced. Completion and extension problems are studied for matrices in this class, under certain positivity constraints. The extension problem is reduced to a standard interpolation problem for Schur matrix valued functions.     

Positive extension and completion problems for a class of structured matrices

A class of matrices, defined by a displacement rank property, is introduced. Completion and extension problems are studied for matrices in this class, under certain positivity constraints. The extension problem is reduced to a standard interpolation problem for Schur matrix valued functions.