A regular matrix pencil sE − A and its rank one perturbations are considered. We determine the sets in C ∪ {∞} which are the eigenvalues of the perturbed pencil. We show that the largest Jordan chains at each eigenvalue of sE − A may disappear and the sum of the length of all destroyed Jordan chains is the number of eigenvalues (counted with multiplicities) which can be placed arbitrarily in C∪...

Recently, we found that the correlation between the eigenvalues of random hermitean matrices exhibits universal behavior. Here we study this universal behavior and develop a diagrammatic approach which enables us to extend our previous work to the case in which the random matrix evolves in time or varies as some external parameters vary. We compute the current-current correlation function, disc...

In this paper, we analyze the distribution of the eigenvalues of glued tridiagonal matrices. Such matrices provide a useful class of test matrix because, despite being unreduced, a glued matrix can have some eigenvalues agreeing to hundreds of decimal places. A glued matrix can be obtained from a direct sum of p copies of an unreduced symmetric tridiagonal matrix T by modifying the junctions, i...

A necessary and sufficient condition is given to a p × q complex matrix X to be an off-diagonal block of an n × n Hermitian matrix C with prescribed eigenvalues (in terms of the eigenvalues of C and singular values of X). The proof depends on some recent breakthroughs in the study of spectral inequalities on the sum of Hermitian matrices by Klyachko and Fulton. Some interesting geometrical prop...

An e$cient method is used to compute the eigenvalues required in a discrete-ordinates solution to a special class of radiative-transfer problems. The basis for this computation is an algorithm for "nding eigenvalues of a matrix that consists of the sum of a diagonal matrix and a rank-one matrix, a form that can arise in a discrete-ordinates solution of some basic transport problems. To illustra...

the Kronecker sum of DD with itself, a total of d times. Using a standard fact about Kronecker sums, if ρ1, . . . , ρN denote the eigenvalues of DD then ρi1 + ρi2 + · · ·+ ρid , i1, . . . , id ∈ {1, . . . , N}, are the eigenvalues of (∆̃) ∆̃. By counting the multiplicity of the zero eigenvalue, we arrive at a nullity for ∆̃ of (k + 1). One can now directly check that each of the polynomials specif...

For a symmetric positive semi-definite diagonally dominant matrix, if its off-diagonal entries and its diagonally dominant parts for all rows (which are defined for a row as the diagonal entry subtracted by the sum of absolute values of off-diagonal entries in that row) are known to a certain relative accuracy, we show that its eigenvalues are known to the same relative accuracy. Specifically, ...

In this paper, we investigate the spectrum of self adjoint operator L defined by := (-1)r d2r/dx2r + A Q(x), where is a operator, Q(x) nuclear in separable Hilbert space, and derive asymptotic formulas for sum eigenvalues L.

The structured pseudospectra of a matrix A are sets of complex numbers that are eigenvalues of matrices X which are near to A and have the same entries as A at a fixed set of places. The sum of multiplicities of the eigenvalues of X inside each connected component of the structured pseudospectra of A does not depend on X. This fact is known, but not so much as it should be. For this reason, we ...