Kh. D. Ikramov

NORMAL MATRICES : AN UPDATE

A list of seventy conditions on an n x n complex matrix A, equivalent to its being normal, published nearly ten years ago by Grone, Johnson, Sa, and Wolkowicz has proved to be very useful. Hoping that, in an extended form, it will be even more helpful, we compile here another list of about twenty conditions. They either have been overlooked by the authors. of the original list or have appeared during the last decade

ON A CONDENSED FORM FOR NORMAL MATRICES UNDER FINITE SEQUENCES OF ELEMENTARY UNITARY SIMILARITIES

Abstract It is generally known that any Hermitian matrix can be reduced to a tridiagonal form by a finite sequence of unitary similarities, namely Householder reflections. Recently A. Bunse-Gerstner and L. Elsner have found a condensed form to which any unitary matrix can be reduced, again by a finite sequence of Householder transformations. This condensed form can be considered as a pentadiagonal or block tridiagonal matrix with some additional zeros inside the band. We describe such a condensed form (or, more precisely, a set of such forms) for general normal matrices, where the number of nonzero elements does not exceed O(n 3 2 ) , n being the order of the normal matrix given. Two approaches to constructing the condensed form are outlined. The first approach is a geometrical Lanczos-type one where we use the so-called generalized Krylov sequences. The second, more constructive approach is an elimination process using Householder reflections. Our condensed form can be thought of as a variable-bandwidth form. An interesting feature of it is that for normal matrices whose spectra lie on algebraic curves of low degree the bandwidth is much smaller.

Matrix pencils: Theory, applications, and numerical methods

This paper surveys nearly all of the publications that have appeared in the last twenty years on the theory of and numerical methods for linear pencils. The survey is divided into the following sections: theory of canonical forms for symmetric and Hermitian pencils and the associated problem of simultaneous reduction of pairs of quadratic forms to canonical form; results on perturbation of characteristic values and deflating subspaces; numerical methods. The survey is self-contained in the sense that it includes the necessary information from the elementary theory of pencils and the theory of perturbations for the common algebraic problem Ax=λx.

A numerical algorithm for solving the matrix equation AX + XTB = C1

An algorithm of the Bartels-Stewart type for solving the matrix equation AX + XTB = C is proposed. By applying the QZ algorithm, the original equation is reduced to an equation of the same type having triangular matrix coefficients A and B. The resulting matrix equation is equivalent to a sequence of low-order systems of linear equations for the entries of the desired solution. Through numerical experiments, the situation where the conditions for unique solvability are “nearly” violated is simulated. The loss of the quality of the computed solution in this situation is analyzed.

On the Distance to the Closest Matrix with Triple Zero Eigenvalue

AbstractThe 2-norm distance from a matrix A to the set

On normal matrices with normal principal submatrices

{\mathcal{M}}

Rational Procedures in the Problem of Common Invariant Subspaces of Two Matrices

of n × n matrices with a zero eigenvalue of multiplicity ≥3 is estimated. If