To solving problems of algebra for two-parameter matrices.~3

V. N. Kublanovskaya ; V. B. Khazanov

Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXI, Tome 359 (2008), pp. 166-207

Résumé

The paper continues the series of papers devoted to surveying and developing methods for solving algebraic problems for two-parameter polynomial and rational matrices of general form. Linearization methods are considered, which allows one to reduce the problem of solving an equation $F(\lambda,\mu)x=0$, with a polynomial two-parameter matrix $F(\lambda,\mu)$, to solving an equation of the form $D(\lambda,\mu)y=0$, where $D(\lambda,\mu)=A(\mu)-\lambda B(\mu)$ is a pencil of polynomial matrices. Consistent pencils and their application to solving spectral problems for the matrix $F(\lambda,\mu)$ are discussed. The notion of reducing subspace is generalized to the case of a pencil of polynomial matrices. An algorithm for transforming a general pencil of polynomial matrices to a quasitriangular pencil is suggested. For a pencil with multiple eigenvalues, algorithms for computing the Jordan chains are developed. Bibl. – 8 titles.