Spectral analysis of matrix polynomials— I. canonical forms and divisors

Abstract

The Jordan normal form for complex matrices is extended to admit “canonical triples” of matrices for monic matrix polynomials and to “standard triples” of operators for monic operator polynomials on finite dimensional linear spaces. These ideas lead to the formulation of canonical, or standard, forms for such polynomials. The inverse problem is also investigated: When do triples of matrices (operators) determine a monic matrix (operator) polynomial for which the triple is canonical (standard)? The canonical and standard forms are very well suited to the study of division and multiplication processes. This is carried out in detail with special emphasis on the question of when matrix (or operator) polynomials have nontrivial divisors which are polynomials of the same kind.