Applied mathematics

Repeated eigenvalues of a matrix with elements polynomially dependent on a parameter

E. A. Kalinina

St. Petersburg State University, 7–9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation

Abstract: Earlier matrices with multiple eigenvalues were considered only for theoretical purposes. However, now such non-generic matrices are also of practical interest because they appear in different problems of quantum mechanics, nuclear physics, optics and dynamic of mechanical systems. A square matrix with elements that are linearly dependent on a parameter is considered in this paper. A method to find the values of the parameter such that the matrix has a repeated eigenvalue is considered. We find the polynomial whose roots are these values of the parameter using only finite number of algebraic operations on the matrix elements. The method can be generalized to matrices with elements algebraically dependent on a parameter. A numerical example of how to find the required parameter values is considered. Refs 16.

Keywords: the Kronecker product, the Leverrier method, the Newton sums.

UDC: 512.643

Received: December 10, 2015
Accepted: February 25, 2016

DOI: 10.21638/11701/spbu10.2016.203

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