Quasi-triangularization of matrix polynomials over arbitrary fields Articles uri icon

publication date

  • May 2023

start page

  • 61

end page

  • 106


  • 665

International Standard Serial Number (ISSN)

  • 0024-3795

Electronic International Standard Serial Number (EISSN)

  • 1873-1856


  • 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).


  • Mathematics


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