Related papers: Eigenvalues and Jordan Forms of Dual Complex Matri…

In this study, we introduce the concept of commutative quaternions and commutative quaternion matrices. Firstly, we give some properties of commutative quaternions and their Hamilton matrices. After that we investigate commutative…

Characterizing the $6\times 6$ complex Hadamard matrices (CHMs) is an open problem in linear algebra and quantum information. In this paper, we investigate the eigenvalues and eigenvectors of CHMs. We show that any $n\times n$ CHM with…

We compute all massive partition functions or characteristic polynomials and their complex eigenvalue correlation functions for non-Hermitean extensions of the symplectic and chiral symplectic ensemble of random matrices. Our results are…

In this paper, we derive new relative perturbation bounds for eigenvectors and eigenvalues for regular quadratic eigenvalue problems of the form $\lambda^2 M x + \lambda C x + K x = 0$, where $M$ and $K$ are nonsingular Hermitian matrices…

Let A(z) be an analytic square matrix and $\lambda_{0}$ an eigenvalue of A(0) of multiplicity m. Then under the generic condition, the characteristic polynomial of A(z) evaluated at $\lambda_{0}$ has a simple zero at z=0, we prove that the…

The nonzero eigenvalues of $AB$ are equal to those of $BA$: an identity that holds as long as the products are square, even when $A,B$ are rectangular. This fact naturally suggests an efficient algorithm for computing eigenvalues and…

Let $\mathbb{K}$ be a field of characteristic different from $2$, and let $M_n(\mathbb{K})$ be the algebra of all $n\times n$ matrices over $\mathbb{K}$. We consider the corresponding special Jordan algebra $\mathcal{A}:=M_n(\mathbb{K})^+$…

We study multiplicative statistics for the eigenvalues of unitarily-invariant Hermitian random matrix models. We consider one-cut regular polynomial potentials and a large class of multiplicative statistics. We show that in the large matrix…

A special class of Jordan algebras over a field $F$ of characteristic zero is considered. Such an algebra consists of an $r$-dimensional subspace of the vector space of all square matrices of a fixed order $n$ over $F$. It contains the…

We consider eigenvalue condition numbers and backward errors for a class of symmetric nonlinear eigenvalue problems with eigenvector nonlinearities. For both of these quantities, we derive explicit and computable expressions that can be…

The general correlation function for the eigenvalues of $p$ complex hermitian matrices coupled in a chain is given as a single determinant. For this we use a slight generalization of a theorem of Dyson.

We describe the structure of all continuous Jordan triple endomorphisms of the set $\mathbb{P}_2$ of all positive definite $2\times 2$ matrices thus completing a recent result of ours. We also mention an application concerning sorts of…

We study the category of representations of $\mathfrak{sl}_{m+2n}$ in positive characteristic, whose p-character is a nilpotent whose Jordan type is the two-row partition (m+n,n). In a previous paper with Anno, we used…

We consider possible non-signaling composites of probabilistic models based on euclidean Jordan algebras. Subject to some reasonable constraints, we show that no such composite exists having the exceptional Jordan algebra as a direct…

We describe two main classes of one-sided trigonometric and hyperbolic Jacobi-type algorithms for computing eigenvalues and eigenvectors of Hermitian matrices. These types of algorithms exhibit significant advantages over many other…

We reformulate the zero-dimensional hermitean one-matrix model as a (nonlocal) collective field theory, for finite~$N$. The Jacobian arising by changing variables from matrix eigenvalues to their density distribution is treated {\it…

In representation theory, the double centraliser property is an important property for a module (bimodule). It plays a fundamental role in many theories. In this paper, we extend this property to complexes in derived categories of finite…

The eigenvector-eigenvalue identity relates the eigenvectors of a Hermitian matrix to its eigenvalues and the eigenvalues of its principal submatrices in which the jth row and column have been removed. We show that one-dimensional arrays of…

We study the right eigenvalue equation for quaternionic and complex linear matrix operators defined in n-dimensional quaternionic vector spaces. For quaternionic linear operators the eigenvalue spectrum consists of n complex values. For…

We study Hadamard variation of eigenvalues of Laplacian with respect to general domain perturbations. We show their existence up to the second order rigorously and characterize the derivatives, using associated eigenvalue problems in finite…