The rank of the difference of matrices with prescribed similarity classes

Let F be a field and let A,B be n × n matrices over I. We study the rank of A' - B' when A and B run over the set of matrices similar to A and B, respectively.     

Sums and products of matrices with prescribed similarity classes

Let F be a field and let A and n × n matrices over F. We study some properties of A^' + B^' and A^'B^', when A^' and B^' run over the sets of the matrices similar to A and B, respectively.     

The eigenvalues of the sum of matrices with prescribed invariant polynomials *

Let A and B be n×n matrices over a field F, and c ₁,…,c_n ∈F. We give a sufficient condition for the existence of matrices A' and B' similar to A and B, respectively, such that A' + B' has eigenvalues c ₁,…,c_n .     

Existence of matrices with prescribed eigenvalues and entries

It is proved improving a previous result of Oliveira that apart from two exceptions there always exists an n×n matrix with arbitarily prescribed 2n-3 entries and spectrum. Moreover, it is shown that the number 2n 3 of prescribed entries cannot be increased.     

The determinant of the sum of two normal matrices with prescribed eigenvalues

Given complex numbers α₁,...,α_n, β₁,...,β_n, what can we say about the determinant of A+B, where A (B) is an n×n normal matrix with eigenvalues α₁,...,α_n (β₁,...,β_n)? Some partial answers are offered to this question.     

On the eigenvalues of matrices with prescribed upper triangular part

Consider a matrix D=[D_i,j] partitioned in submatrices D_i,j where the submatrices D_i,iare square. This paper studies the possible eigenvalues of D, when the blocks D_i,j, with i≤j, are fixed, and the other blocks vary. The result obtained generalizes twotheorems already known.     

On some classes of structured matrices with algebraic trigonometric eigenvalues

Diagonal plus semiseparable matrices are constructed, the eigenvalues of which are algebraic numbers expressed by simple closed trigonometric formulas.     

Bounds on eigenvalues of the Hadamard product and the Fan product of matrices

We prove an upper bound for the spectral radius of the Hadamard product of nonnegative matrices and a lower bound for the minimum eigenvalue of the Fan product of M-matrices. These improve two existing results.     

Estimates for the eigenvalues of the Jordan product of Hermitian matrices

Given Hermitian matrices A and B, Professor Taussky-Todd posed the problem of estimating the eigenvalues of their Jordan product AB+BA. Here we establish bounds for all the eigenvalues of the Jordan product when both A and B are positive definite. At the same time we give a more straightforward proof and an improvement of estimates given by D. W. Nicholson for the smallest eigenvalue.     

On limit theorem for the eigenvalues of product of two random matrices

The existence of limiting spectral distribution (LSD) of the product of two random matrices is proved. One of the random matrices is a sample covariance matrix and the other is an arbitrary Hermitian matrix. Specially, the density function of LSD of S_nW_n is established, where S_n is a sample covariance matrix and W_n is Wigner matrix.     

A limit theorem for the eigenvalues of product of two random matrices

The existence of limit spectral distribution of the product of two independent random matrices is proved when the number of variables tends to infinity. One of the above matrices is the Wishart matrix and the other is a symmetric nonnegative definite matrix.     

On similarity classes of second order matrices with zero trace over the ring of integers

In this paper we consider the problem of similarity of matrices over integers and describe canonical representatives of the similarity classes of 2 × 2 matrices with zero trace and determinant equal an odd power of a prime number.