  The purpose of this manuscript is to derive the Hoffman-Wielandt inequality
and its most general form for quaternion matrices. Diagonalizabilty 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 $\frac{1}{2} < |\lambda_0| < 2$.
