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

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 $Q(\lambda )$ 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 $P(\lambda )$ is a quaternion matrix polynomial with unitary coefficients, then any right eigenvalue ${\lambda }_0$ of $P(\lambda )$ lies in the annular region $\frac{1}{2}<\mathrm{\mid }{\lambda }_0\mathrm{\mid }<2$.