On a certain class of spectral characteristic of matrices (Q1920817)

It is well known that finding the eigenvalues of nonsymmetric matrices by computer can lead to essential errors. Therefore, in studying the location of the spectrum of a matrix \(A\) in the complex plane or studying the behavior of solutions to a system of differential equations (1) \({dy\over dt}=Ay\), the natural problem arises that consists in finding numerical characteristics of the matrix \(A\) whose use would allow one to answer the posed questions with guaranteed accuracy. Using these characteristics, it becomes possible to work out a series of algorithms for solving certain problems of linear algebra. The main aim of the present article is to indicate a collection of numerical characteristics whose use makes it possible to distinguish those matrices whose spectra belong to the closed semiplane \(\{\text{Re} \lambda\leq 0\}\); moreover, a principal opportunity is revealed to carry out the calculations by computer. The use of such characteristics enables us to solve the problems of location of the spectrum of a matrix on a straight line, in a strip, or in a convex polygon of the complex plane. By the way, we solve the problem of finding a numerical criterion for the Lyapunov stability of solutions to system (1) which can be tested by computer. Also, we present some stability theorems in integral form.