Relations between ranks of matrix polynomials

TL;DR

This paper establishes a fundamental relationship between the ranks of matrix polynomials and their greatest common divisor and least common multiple, providing a new perspective on matrix polynomial rank properties.

Contribution

It introduces a novel rank relation involving the gcd and lcm of matrix polynomials, with multiple applications to existing problems.

Findings

Sum of ranks of two matrix polynomials equals the sum of ranks of matrices from their gcd and lcm.

The result applies to various classical and recent problems in matrix polynomial theory.

Provides a unifying framework for understanding rank relations in matrix polynomial algebra.

Abstract

We show that the sum of ranks of two matrix polynomials is the same as the sum of the rank of the matrix obtained by applying the greatest common divisor of the polynomials, with the rank of the matrix obtained by applying the lowest common multiple of the polynomials. Many applications, for older or more recent problems, of this result are obtained.