DETERMINING SOME CHARACTERISTICS OF ONE-PARAMETRIC KRONECKER MATRICES

Abstract

Methods for determining some characteristics of one-parameter Kronecker matrices are presented, in particular, methods for determining traces, determinants, eigenvalues-functions and singular values-functions. In all cases, differential transformations act as the main mathematical apparatus, which make it possible to obtain simple recurrent computational procedures that are easily implemented by means of modern information technologies.

Well-known matrices are used in this work. To determine the trace of the corresponding one-parameter Kronecker matrix, a direct analytical method, an analytical method for determining the one-parameter Kronecker matrix itself, and also a method based on differential transformations are used. On the basis of the results obtained, an algorithm for determining the traces of one-parameter Kronecker matrices is proposed.

With the same known matrices, to determine the determinant of the corresponding one-parameter Kronecker matrix, the well-known approach for autonomous matrices, the method of finding the determinant by definition, and the method based on differential transformations are applied. On the basis of the results obtained, an algorithm is proposed for determining the determinants of one-parameter Kronecker matrices.

To determine the eigenvalues-functions of the corresponding one-patametric Kronecker matrix for the same matrices considered, the well-known approach for autonomous matrices is also applied. On the basis of the results obtained, an algorithm for determining the eigenvalues-functions of one-parameter Kronecker matrices is proposed.

To determine the singular values-functions of one-parameter Kronecker matrices, an algorithm based on the use of differential transformations taking into account the number of nonzero singular values-functions is also proposed.

All algorithms are invariants with respect to the sizes of the considered matrices and their elements. The only limitation is the smoothness of the elements of these matrices at the corresponding centers of approximation.