Abstract. The main result of this paper is the extension of the Schur-Horn Theorem to infinite sequences: For two nonincreasing nonsummable sequences ξ and η that converge to 0, there exists a compact operator A with eigenvalue list η and diagonal sequence ξ if and only if Pn j=1 ξj ≤ Pn j=1 ηj for every n if and only if ξ = Qη for some orthostochastic matrix Q. The similar result requiring equ...

We generalize the classical notion of majorization in Rn to a majorization order for functions defined on a partially ordered set P . In this generalization we use inequalities for partial sums associated with ideals in P . Basic properties are established, including connections to classical majorization. Moreover, we investigate transfers (given by doubly stochastic matrices), complexity issue...

We estabish an analog of the Poincare-Cauchy separation theorem for normal matrices in terms of majorization. Moreover, we present a solution to the inverse spectral problem (Borgtype result) for a normal matrix. Using this result we essentially generalize and complement the known Gauss–Lucas theorem on the geometry of the roots of a complex polynomial and of its derivative. In turn the last re...

Abstract Rank-constrained spatial covariance matrix estimation (RCSCME) is a blind speech extraction method utilized under the condition that one-directional target and diffuse background noise are mixed. In this paper, we propose new model extension of RCSCME. RCSCME simultaneously conducts both deficient rank-1 component complementation matrix, which incompletely estimated by preprocessing me...

The Rayleigh-Ritz method finds the stationary values, called Ritz values, of the Rayleigh quotient on a given trial subspace as approximations to eigenvalues of a Hermitian operator A. If the trial subspace is A-invariant, the Ritz values are exactly some of the eigenvalues of A. Given two subspaces X and Y of the same finite dimension, such that X is A-invariant, the absolute changes in the Ri...

We introduce a partial order, variance majorization, on R, which is analogous to the majorization order. A new class of monotonicity inequalities, based on variance majorization and analogous to Schur convexity, is developed.

By using diagonalizable matrix decomposition and majorization inequalities, we propose new trace bounds for the product of two real square matrices in which one is diagonalizable. These bounds improve and extend the previous results. Furthermore, we give some trace bounds for the solution of the algebraic Riccati equations, which improve some of the previous results under certain conditions. Fi...

A positive map between Euclidean Jordan algebras is a (symmetric cone) order preserving linear map. We show that the norm of such a map is attained at the unit element, thus obtaining an analog of the operator/matrix theoretic Russo-Dye theorem. A doubly stochastic map between Euclidean Jordan algebras is a positive, unital, and trace preserving map. We relate such maps to Jordan algebra automo...

Given an approximate invariant subspace we discuss the effectiveness of majorization bounds for assessing the accuracy of the resulting Rayleigh-Ritz approximations to eigenvalues of Hermitian matrices. We derive a slightly stronger result than previously for the approximation of k extreme eigenvalues, and examine some advantages of these majorization bounds compared with classical bounds. From...

We study the Majorization arrow in a big class of quantum adiabatic algorithms. In a quantum adiabatic algorithm, the ground state of the Hamiltonian is a guide state around which the actual state evolves. We prove that for any algorithm of this class, step-by-step majorization of the guide state holds perfectly. We also show that step-by-step majorization of the actual state appears if the run...