On the eigenvalues of principal submatrices of J-normal matrices

The problem of the existence of a J-normal matrix A when its spectrum and the spectrum of some of its (n-1)×(n-1) principal submatrices are prescribed is analyzed. The case of 3×3 matrices is particularly investigated. The results here obtained in the framework of indefinite inner product spaces are in the spirit of those due to Nylen, Tam and Uhlig.     

Total nonpositivity of nonsingular matrices

In this paper, nonsingular totally nonpositive matrices are studied and new characterizations are provided in terms of the signs of minors with consecutive initial rows or consecutive initial columns. These characterizations extend an existing characterization that uses some restrictive hypotheses.     

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.     

Extension to maximal semidefinite invariant subspaces for hyponormal matrices in indefinite inner products

It is proved that under certain essential additional hypotheses, a nonpositive invariant subspace of a hyponormal matrix admits an extension to a maximal nonpositive subspace which is invariant for both the matrix and its adjoint. Nonpositivity of subspaces and the hyponormal property of the matrix are understood in the sense of a nondegenerate inner product in a finite dimensional complex vector space. The obtained theorem combines and extends several previously known results. A Pontryagin space formulation, with essentially the same proof, is offered as well.     

Suborthogonality and orthocentricity of matrices

We present some results on submatrices of orthogonal and unitary matrices and their relation to so called orthocentric matrices. These are then completely characterized.     

A min-max theorem for complex symmetric matrices

We optimize the form Re x^tTx to obtain the singular values of a complex symmetric matrix T. We prove that for ,

     

On zero-pattern invariant properties of structured matrices

It is interesting that inverse M-matrices are zero-pattern (power) invariant. The main contribution of the present work is that we characterize some structured matrices that are zero-pattern (power) invariant. Consequently, we provide necessary and sufficient conditions for these structured matrices to be inverse M-matrices. In particular, to check if a given circulant or symmetric Toeplitz matrix is an inverse M-matrix, we only need to consider its pattern structure and verify that one of its principal submatrices is an inverse M-matrix.     

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.     

Hyponormal and strongly hyponormal matrices in inner product spaces

Complex matrices that are structured with respect to a possibly degenerate indefinite inner product are studied. Based on earlier works on normal matrices, the notions of hyponormal and strongly hyponormal matrices are introduced. A full characterization of such matrices is given and it is shown how those matrices are related to different concepts of normal matrices in degenerate inner product spaces. Finally, the existence of invariant semidefinite subspaces for strongly hyponormal matrices is discussed.     

Loewner matrices and operator convexity

Let f be a function from \({\mathbb{R}_{+}}\) into itself. A classic theorem of K. Löwner says that f is operator monotone if and only if all matrices of the form \({\left [\frac{f(p_i) - f(p_j)}{p_i-p_j}\right ]_{\vphantom {X_{X_1}}}}\) are positive semidefinite. We show that f is operator convex if and only if all such matrices are conditionally negative definite and that f (t) = t g(t) for some operator convex function g if and only if these matrices are conditionally positive definite. Elementary proofs are given for the most interesting special cases f (t) = t ^r , and f (t) = t log t. Several consequences are derived.     

On operator inequalities of some relative operator entropies

In our recent paper, we introduced the notions of relative operator (α,β)$(\alpha ,\beta )$-entropy and Tsallis relative operator (α,β)$(\alpha ,\beta )$-entropy as a parameter extensions of relative operator entropy and Tsallis relative operator entropy. In this paper, we give upper and lower bounds of these new notions according to operator (α,β)$(\alpha ,\beta )$-geometric mean introduced in Nikoufar et al. (2013) [14].     

The simplest proof of Lieb concavity theorem

In this paper, by applying jointly concavity and jointly convexity of generalized perspective of some elementary functions, we give the simplest proof of the well-known Lieb concavity theorem and Ando convexity theorem.     

Equalities and inequalities for inertias of hermitian matrices with applications

The inertia of a Hermitian matrix is defined to be a triplet composed of the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we show some basic formulas for inertias of 2×2 block Hermitian matrices. From these formulas, we derive various equalities and inequalities for inertias of sums, parallel sums, products of Hermitian matrices, submatrices in block Hermitian matrices, differences of outer inverses of Hermitian matrices. As applications, we derive the extremal inertias of the linear matrix expression A-BXB^∗ with respect to a variable Hermitian matrix X. In addition, we give some results on the extremal inertias of Hermitian solutions to the matrix equation AX=B, as well as the extremal inertias of a partial block Hermitian matrix.     

An algorithm for determining copositive matrices

In this paper, we present an algorithm of simple exponential growth called COPOMATRIX for determining the copositivity of a real symmetric matrix. The core of this algorithm is a decomposition theorem, which is used to deal with simplicial subdivision of on the standard simplex Δ_m, where each component of the vector β is −1, 0 or 1.     

On three Krein extension problems and some generalizations

Three basic extension problems which were initiated by M. G. Krein are discussed and further developed. Connections with interpolation problems in the Carathéodory class are explained. Some tangential and bitangential versions are considered. Full characterizations of the classes of resolvent matrices for these problems are given and formulas for the resolvent matrices of left tangential problems are obtained using reproducing kernel Hilbert space methods.Dedicated to the memory of M. G. Krein, a beacon for us both.The authors wish to acknowledge the partial support of the Israel-Ukraine Exchange Program. D. Z. Arov also wishes to thank the Weizmann Institute of Science for partial support and hospitality; H. Dym wishes to thank Renee and Jay Weiss for endowing the chair which supports his research.     

Idempotents of Fourier multiplier algebra

We prove that if the indicator-function1 _E of a measurable setE is a Fourier multiplier in the spaceE ^p () for somep2 thenE is an open set (up to a set of measure zero).     

Alexandrov’s inequality and conjectures on some Toeplitz matrices

We study determinant inequalities for certain Toeplitz-like matrices over C. For fixed n and N ? 1, let Q be the n × (n + N − 1) zero-one Toeplitz matrix with Q_ij = 1 for 0 ? j − i ? N − 1 and Q_ij = 0 otherwise. We prove that det(QQ^∗) is the minimum of det(RR^∗) over all complex matrices R with the same dimensions as Q satisfying ∣R_ij∣ ? 1 whenever Q_ij = 1 and R_ij = 0 otherwise. Although R has a Toeplitz-like band structure, it is not required to be actually Toeplitz. Our proof involves Alexandrov’s inequality for polarized determinants and its generalizations. This problem is motivated by Littlewood’s conjecture on the minimum 1-norm of N-term exponential sums on the unit circle. We also discuss polarized Bazin-Reiss-Picquet identities, some connections with k-tree enumeration, and analogous conjectured inequalities for the elementary symmetric functions of QQ^∗.     

On the spectrum of a matrix pencil and two-side infinite periodic Jacobi matrices

The spectrum and the Jordan structure of a matrix pencilA _z =z ^–1 B+C+zB ^T has been considered. The results have been applied to investigation of the spectrum of two-side infinite periodic Jacobi matrices.     

The courant-fischer theorem and the spectrum of selfadjoint block band Toeplitz operators

We show that ifT(F) is a selfadjoint block Toeplitz operator generated by a trigonometric matrix polynomialF, then the spectrum ofT(F) as well as the limiting set (F) of the eigenvalues of the truncationsT _n (F) is the union of a finite collection of segments (the spectral range ofF) and at most a finite set of points for which we give an upper bound.     

On rank invariance of generalized Schwarz-Pick-Potapov block matrices of matrix-valued Carathéodory functions

In view of a multiple Nevanlinna-Pick interpolation problem, we study the rank of generalized Schwarz-Pick-Potapov block matrices of matrix-valued Carathéodory functions. Those matrices are determined by the values of a Carathéodory function and the values of its derivatives up to a certain order. We derive statements on rank invariance of such generalized Schwarz-Pick-Potapov block matrices. These results are applied to describe the case of exactly one solution for the finite multiple Nevanlinna-Pick interpolation problem and to discuss matrix-valued Carathéodory functions with the highest degree of degeneracy.