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For given real or complex $m \times n$ data matrices $X$, $Y$, we investigate when there is a matrix $A$ such that $AX = Y$, and $A$ is invertible, Hermitian, positive (semi)definite, unitary, an orthogonal projection, a reflection, complex…
Given an n x n matrix over the ring of differential polynomials F(t)[\D;\delta], we show how to compute the Hermite form H of A, and a unimodular matrix U such that UA=H. The algorithm requires a polynomial number of operations in terms of…
For certain tame abelian covers of arithmetic surfaces X/Y we obtain striking formulas, involving a quadratic form derived from intersection numbers, for the equivariant Euler characteristics of both the canonical sheaf !X/Y and also its…
Consider a matrix function f defined for Hermitian matrices. The purpose of this paper is two-fold. We derive new results for the absolute structured condition number of the matrix function and we derive new bounds for the perturbation…
The structure of weighting coefficient matrices of Harmonic Differential Quadrature (HDQ) is found to be either centrosymmetric or skew centrosymmetric depending on the order of the corresponding derivatives. The properties of both matrices…
We prove quadratic eigenvalue perturbation bounds for generalized Hermitian eigenvalue problems. The bounds are proportional to the square of the norm of the perturbation matrices divided by the gap between the spectrums. Using the results…
Dual quaternions can represent rigid body motion in 3D spaces, and have found wide applications in robotics, 3D motion modelling and control, and computer graphics. In this paper, we introduce three different right linear independency for a…
Parametric linear systems are linear systems of equations in which some symbolic parameters, that is, symbols that are not considered to be candidates for elimination or solution in the course of analyzing the problem, appear in the…
Given a square, nonsingular matrix of univariate polynomials $\mathbf{F} \in \mathbb{K}[x]^{n \times n}$ over a field $\mathbb{K}$, we give a fast, deterministic algorithm for finding the Hermite normal form of $\mathbf{F}$ with complexity…
For solving the continuous Sylvester equation, a class of the multiplicative splitting iteration method is presented. We consider two symmetric positive definite splittings for each coefficient matrix of the continuous Sylvester equations…
We construct a quantum-inspired classical algorithm for computing the permanent of Hermitian positive semidefinite matrices, by exploiting a connection between these mathematical structures and the boson sampling model. Specifically, the…
The problem of expressing a multivariate polynomial as the determinant of a monic (definite) symmetric or Hermitian linear matrix polynomial (LMP) has drawn a huge amount of attention due to its connection with optimization problems. In…
It is known that a $2\times 2$ quaternionic matrix has one, two or an infinite number of left eigenvalues, but the available algebraic proofs are difficult to generalize to higher orders. In this paper a different point of view is adopted…
A sum rule relative to a reference measure on R is a relationship between the reversed Kullback-Leibler divergence of a positive measure on R and some non-linear functional built on spectral elements related to this measure (see for example…
We compute two parametric determinants in which rows and columns are indexed by compositions, where in one determinant the entries are products of binomial coefficients, while in the other the entries are products of powers. These results…
Commutative analogues of Clifford algebras are algebras defined in the same way as Clifford algebras except that their generators commute with each other, in contrast to Clifford algebras in which the generators anticommute. In this paper,…
We derive the loop equations for the one Hermitian matrix model in any dimension. These are a consequence of the Schwinger-Dyson equations of the model. Moreover we show that in leading order of large $N$ the loop equations form a closed…
Assume that the eigenvalues of a finite hermitian linear operator have been deduced accurately but the linear operator itself could not be determined with precision. Given a set of eigenvalues $\lambda$ and a hermitian matrix $M$, this…
This note deals with two topics of linear algebra. We give a simple and short proof of the multiplicative property of the determinant and provide a constructive formula for rotations. The derivation of the rotation matrix relies on simple…
Canonical matrices of (a) bilinear and sesquilinear forms, (b) pairs of forms, in which every form is symmetric or skew-symmetric, and (c) pairs of Hermitian forms are given over finite fields of characteristic not 2 and over finite…