The Bezoutian and the eigenvalue-separation problem for matrix polynomials

A generalized Bezout matrix for a pair of matrix polynomials is studied and, in particular, the structure of its kernel is described and the relations to the greatest common divisor of the given matrix polynomials are presented. The classical root-separation problems of Hermite, Routh-Hurwitz and Schur-Cohn are solved for matrix polynomials in terms of this Bezout matrix. The eigenvalue-separation results are also expressed in terms of Hankel matrices whose entries are Markov parameters of rational matrix function. Some applications of Jacobi's method to these problems are pointed out.

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