Ikramov Kh.D.

We discuss how to get a normal matrix from a binormal one and, conversely, how to get a binormal matrix from a normal one via the right multiplication on a suitable unitary matrix. Let N be a normal matrix badly conditioned with respect to inversion, that is, having a large condition number cond2N. We show that, among the binormal matrices B that can be obtained from N, there is a matrix whose eigenvalues have individual condition numbers of order (cond2N)1/2.

Similarity transformations are the main part of matrix theory, which studies numerous classes of special matrices. Accordingly, there are many ways of describing such classes. In most cases, one can verify whether a matrix belongs to the required class by a rational calculation, that is, by a finite algorithm using only arithmetical operations.

The matrix equation XAX = AXA is called the equation of the Young–Baxter type. We examine this equation for matrices of order 2 under the assumption that A is a nonsingular matrix, and we are only interested in the nonsingular solutions. Using a unified rule, one can associate with every such solution a matrix commuting with A. In other words, this matrix is an element of the centralizer MA of A. No obvious reasons exist for two distinct solutions X1 and X2 to generate the same element of MA. Nevertheless, all the solutions (and there are infinitely many of them) generate one and the same matrix in the centralizer. We give an explanation of this surprising fact.

Let A and B be matrices of order n that are direct sums of nilpotent Jordan blocks. Suppose that A and B are not just different arrangements of the same blocks but, rather, they differ in the sizes of the blocks. It is shown that, in this case, A and B cannot be congruent. This result can be regarded as a new proof of the uniqueness of the singular part in the Horn–Sergeichuk canonical form of a singular matrix.