Related papers: New properties for certain positive semidefinite m…
This paper examines the properties of real symmetric square matrices with a constant value for the main diagonal elements and another constant value for all off-diagonal elements. This matrix form is a simple subclass of circulant matrices,…
We define geometric matrix midranges for positive definite Hermitian matrices and study the midrange problem from a number of perspectives. Special attention is given to the midrange of two positive definite matrices before considering the…
Semidefinite and sum-of-squares (SOS) optimization are fundamental computational tools in many areas, including linear and nonlinear systems theory. However, the scale of problems that can be addressed reliably and efficiently is still…
Let $\mathbb{A}=\left( \begin{array}{cc} A & 0 \\ 0 & A \\ \end{array} \right)$ be the $2\times2$ diagonal operator matrix determined by a positive bounded operator $A$. For semi-Hilbertian operators $X$ and $Y$, we first show that…
Efficient solution of the lowest eigenmodes is studied for a family of related eigenvalue problems with common $2\times 2$ block structure. It is assumed that the upper diagonal block varies between different versions while the lower…
This paper establishes new upper bounds for the right eigenvalues of monic matrix polynomials over the quaternion division algebra. The noncommutative nature of quaternion multiplication presents fundamental challenges in eigenvalue…
Given a block triangular matrix $M$ over a noncommutative ring with invertible diagonal blocks, this work gives two new representations of its inverse $M^{-1}$. Each block element of $M^{-1}$ is explicitly expressed via a quasideterminant…
Estimation of top singular values is one of the widely used techniques and one of the intensively researched problems in Numerical Linear Algebra and Data Science. We consider here two general questions related to this problem: How top…
We review some of the significant generalizations and applications of the celebrated Douglas theorem on the equivalence of factorization, range inclusion, and majorization of operators. We then apply it to find a characterization of the…
In this paper we discuss some relations between the eigenvalues and the diagonal entries of Hermitian matrices.
We present an Oppenheim type determinantal inequality for positive definite block matrices. Recently, Lin [Linear Algebra Appl. 452 (2014) 1--6] proved a remarkable extension of Oppenheim type inequality for block matrices, which solved a…
We obtain a complete characterization of the $2\times 2$ symplectic matrices having an infinite number of left eigenvalues. Previously, we give a new proof of a result from Huang and So about the number of eigenvalues of a quaternionic…
Let H be a positive semidefinite matrix partitioned into Hermitian blocks. Then, up to a direct sum operation, H is the average of matrices isometrically congruent to its partial trace. A few corollaries are given, related to important…
We present a new approach to compute selected eigenvalues and eigenvectors of the two-parameter eigenvalue problem. Our method requires computing generalized eigenvalue problems of the same size as the matrices of the initial two-parameter…
Recently, the eigenvalue problems formulated with symmetric positive definite bilinear forms have been well investigated with the aim of explicit bounds for the eigenvalues. In this paper, the existing theorems for bounding eigenvalues are…
We show 2 matrices that have identical eigenvalues but different eigenfunctions. This shows that in obtaining two body nuclear matrix elements empirically, it is not sufficient to consider only energy levels. Other quantities like…
We investigate determinantal varieties for symmetric matrices that have zero blocks along the main diagonal. In theoretical physics, these arise as Gram matrices for kinematic variables in quantum field theories. We study the ideals of…
In this paper, we introduce a set of block factor-width-two matrices, which is a generalisation of factor-width-two matrices and is a subset of positive semidefinite matrices. The set of block factor-width-two matrices is a proper cone and…
Several subadditivity results and conjectures are given for matrices (or operators), block-matrices, concave functions and norms.
The dimensions of sets of matrices of various types, with specified eigenvalue multiplicities, are determined. The dimensions of the sets of matrices with given Jordan form and with given singular value multiplicities are also found. Each…