Quasi-triangularization of matrix polynomials over arbitrary fields

In [19], Taslaman, Tisseur, and Zaballa show that any regular matrix polynomial P (λ)over an algebraically closed ﬁeld is spectrally equivalent to a triangular matrix polynomial of the same degree. When P (λ)is real and regular, they also show that there is a real quasi-triangular matrix polynomial of the same degree that is spectrally equivalent to P (λ), in which the diagonal blocks are of size at most 2 × 2. This paper generalizes these results to regular matrix polynomials P (λ)over arbitrary ﬁelds F, showing that any such P (λ) can be quasi-triangularized to a spectrally equivalent matrix polynomial over F of the same degree, in which the largest diagonal block size is bounded by the highest degree appearing among all of the F-irreducible factors in the Smith form for P (λ).

matrix polynomials; elementary divisors; inverse problem; triangularization; arbitrary fields; majorization; homogeneous partitioning

Quasi-triangularization of matrix polynomials over arbitrary fields Articles