The Hoffman-Wielandt inequality for quaternion matrices and quaternion matrix polynomials

TL;DR

This work addresses spectral variation for quaternion matrices and quaternion matrix polynomials by developing a quaternion analogue of the Hoffman-Wielandt inequality using the complex adjoint representation $\chi_A$ and block companion matrices. It proves a sharp bound for standard right eigenvalues of normal (and certain diagonalizable) quaternion matrices and extends the result to block companion matrices of quaternion matrix polynomials under specific coefficient conditions, including diagonalizability results. A key contribution is a generalized perturbation bound for the standard eigenvalues of block companion matrices of quadratic and linear quaternion matrix polynomials, with precise conditions on the coefficient matrices. The paper also establishes an annular bound $\tfrac{1}{2}<|\lambda_0|<2$ for right eigenvalues of quaternion matrix polynomials when the coefficients are unitary, linking quaternion and complex adjoint frameworks and posing open questions about left eigenvalues and broader polynomial classes.

Abstract

The purpose of this paper is to derive the Hoffman-Wielandt inequality and its generalization for quaternion matrices. Diagonalizability of the block companion matrix of certain quadratic (linear) quaternion matrix polynomials is brought out. As a consequence, we prove that if $Q(λ)$ is another quadratic (linear) quaternion matrix polynomial, then under certain conditions on the coefficients, a generalization of the Hoffman-Wielandt inequality for their corresponding block companion matrices holds. We also prove that if $P(λ)$ is a quaternion matrix polynomial with unitary coefficients, then any right eigenvalue $λ_0$ of $P(λ)$ lies in the annular region $\frac{1}{2} < |λ_0| < 2$.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Remark 2.2
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Lemma 3.3
• Example 3.4