Related papers: Smaller Gershgorin disks for multiple eigenvalues …
Conjugation covariants of matrices are applied to study the real algebraic variety consisting of complex Hermitian matrices with a bounded number of distinct eigenvalues. A minimal generating system of the vanishing ideal of degenerate…
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…
Let M be an arbitrary Hermitian matrix of order n, and k be a positive integer less than or equal to n. We show that if k is large, the distribution of eigenvalues on the real line is almost the same for almost all principal submatrices of…
This paper is concerned with computations of a few smaller eigenvalues (in absolute value) of a large extremely ill-conditioned matrix. It is shown that smaller eigenvalues can be accurately computed for a diagonally dominant matrix or a…
We prove that semialgebraic sets of rectangular matrices of a fixed rank, of skew-symmetric matrices of a fixed rank and of real symmetric matrices whose eigenvalues have prescribed multiplicities are minimal submanifolds of the space of…
A matrix is well separated if all its Gershgorin circles are away from the unit circle and they are separated from each other. In this article, the region of relative errors in the eigenvalues is obtained as a quadratic oval for non…
Two types of eigenvalue continuity are commonly used in the literature. However, their meanings and the conditions under which continuities are used are not always stated clearly. This can lead to some confusion and needs to be addressed.…
Statistical properties of non--symmetric real random matrices of size $M$, obtained as truncations of random orthogonal $N\times N$ matrices are investigated. We derive an exact formula for the density of eigenvalues which consists of two…
We rederive in a simplified version the Lehmann-Sommers eigenvalue distribution for the Gaussian ensemble of asymmetric real matrices, invariant under real orthogonal transformations, as a basis for a detailed derivation of a Pfaffian…
The distributions of the smallest and largest eigenvalues for the matrix product $Z^\dagger Z$, where $Z$ is an $n \times m$ complex Gaussian matrix with correlations both along rows and down columns, are expressed as $m \times m$…
A new approach to solving eigenvalue optimization problems for large structured matrices is proposed and studied. The class of optimization problems considered is related to computing structured pseudospectra and their extremal points, and…
We obtain generalisations of some inequalities for positive unital linear maps on matrix algebra. This also provides several positive semidefinite matrices and we get some old and new inequalities involving the eigenvalues of a Hermitian…
The affine inverse eigenvalue problem consists of identifying a real symmetric matrix with a prescribed set of eigenvalues in an affine space. Due to its ubiquity in applications, various instances of the problem have been widely studied in…
Pseudo-hermitian matrices are matrices hermitian with respect to an indefinite metric. They can be thought of as the truncation of pseudo-hermitian operators, defined over some Krein space, together with the associated metric, to a finite…
In this note, we present a systematic method to explicitly compute the determinants and inverses for some generalized Hilbert matrices associated with orthogonal systems with explicit representations. We expressed the determinant, the…
The product of a Hermitian matrix and a positive semidefinite matrix has only real eigenvalues. We present bounds for sums of eigenvalues of such a product.
We gather several results on the eigenvalues of the spatial sign covariance matrix of an elliptical distribution. It is shown that the eigenvalues are a one-to-one function of the eigenvalues of the shape matrix and that they are closer…
For given k distinct complex conjugate pairs, l distinct real numbers, and a given graph G on 2k+l vertices with a matching of size at least k, we will show that there is a real matrix whose eigenvalues are the given numbers and its graph…
The statistical properties of trajectories of eigenvalues of Gaussian complex matrices whose Hermitian condition is progressively broken are investigated. It is shown how the ordering on the real axis of the real eigenvalues is reflected in…
In matrix theory and numerical analysis there are two very famous and important results. One is Gersgorin circle theorem, the other is strictly diagonally dominant theorem. They have important application and research value, and have been…