On the Right Eigenvalues of the Quaternionic Matrix Polynomials

TL;DR

This paper develops new upper bounds for the right eigenvalues of quaternionic matrix polynomials, addressing noncommutativity challenges and improving eigenvalue estimates through spectral norm inequalities.

Contribution

It introduces sharper bounds for quaternionic polynomial eigenvalues using spectral norm inequalities and matrix structure analysis.

Findings

Derived progressively sharper bounds for right eigenvalues

Connected eigenvalue bounds to zeros of quaternionic polynomials

Addressed fundamental challenges due to quaternion noncommutativity

Abstract

This paper establishes new upper bounds for the right eigenvalues of monic matrix polynomials over the quaternion division algebra. The noncommutative nature of quaternion multiplication presents fundamental challenges in eigenvalue analysis, distinguishing this problem from the classical complex case. We use spectral norm inequalities for partitioned quaternionic matrices and apply them to quaternionic block matrices associated with monic matrix polynomials. By analyzing the structure of powers of these companion matrices we derive progressively sharper bounds for the right eigenvalues. Consequently, these bounds give bounds for the zeros of quaternionic polynomials.