We describe a new proof of the well-known Lyapunov's matrix inequality about the location of the eigenvalues of a matrix in some region of the complex plane. The proof makes use of standard facts from quadratic and semi-deenite programming. Links are established between the Lyapunov matrix, rank-one LMIs and the Lagrange multiplier arising in duality theory.

This paper specifies some conditions as to when an integer m is locally represented by a positive definite diagonal integer-matrix ternary quadratic form Q at a prime p. We use quadratic Gauss sums and a version of Hensel’s Lemma to count how many solutions there are to the equivalence Q(~x) ≡ m (mod p) for any k ≥ 0. Given that m is coprime to the determinant of the Hessian matrix of Q, we can...

In this paper we will look at two results in which a special para-Hermitian matrix polynomial appears in linear quadratic systems theory. The first result constitutes the first step in a dissipativity check. The second result shows that dissipativity is equivalent to the solvability of the infinitehorizon linear quadratic optimal control problem and that its solutions are given by the behavior ...

Pseudospectra associated with the standard and generalized eigenvalue problems have been widely investigated in recent years. We extend the usual definitions in two respects, by treating the polynomial eigenvalue problem and by allowing structured perturbations of a type arising in control theory. We explore connections between structured pseudospectra, structured backward errors, and structure...

 This paper present unequal-sized facilities layout solutions generated by a genetic search program named LADEGA (Layout Design using a Genetic Algorithm). The generalized quadratic assignment problem requiring pre-determined distance and material flow matrices as the input data and the continuous plane model employing a dynamic distance measure and a material flow matrix are discussed. Computa...

The paper investigates the problem of fitting protein complexes into electron density maps. They are represented by high-resolution cryoEM density maps converted into overlapping matrices and partly show a structure of a complex. The general purpose is to define positions of all proteins inside it. This problem is known to be NP-hard, since it lays in the field of combinatorial optimization ove...

This document contains the appendices for the paper “The Quadratic-Chi Histogram Distance Family” [1], proofs and additional results. In section 2 we prove that all Quadratic-Chi histogram distances are continuous. In section 3 we prove that EMD, ÊMD and all Quadratic-Chi histogram distances are Similarity-Matrix-QuantizationInvariant. In section 4 we present additional shape classification res...

Quadratic Poisson tensors of the Dufour-Haraki classification read as a sum of an r-matrix induced structure twisted by a (small) compatible exact quadratic tensor. An appropriate bigrading of the space of formal Poisson cochains then leads to a vertically positive double complex. The associated spectral sequence allows to compute the Poisson-Lichnerowicz cohomology of the considered tensors. W...

Consider the quadratic form β = y(YY + ρI)y where ρ is a positive number, where y is a random vector and Y is a N × K random matrix both having independent elements with different variances, and where y and Y are independent. Such quadratic forms represent the Signal to Noise Ratio at the output of the linear Wiener receiver for multi dimensional signals frequently encountered in wireless commu...

In this paper, we first give the maximal and minimal values of the ranks and inertias of the quadratic matrix functions q1(X) = Q1 − XP1X and q2(X) = Q2 − XP2X subject to a consistent matrix equation AX = B, where Q1, Q2, P1 and P2 are Hermitian matrices. As applications, we derive necessary and sufficient conditions for the solution of AX = B to satisfy the quadratic equality XP1X ∗ = Q1 and X...