  It is well known that if a matrix $A\in\mathbb C^{n\times n}$ solves the
matrix equation $f(A,A^H)=0$, where $f(x, y)$ is a linear bivariate polynomial,
then $A$ is normal; $A$ and $A^H$ can be simultaneously reduced in a finite
number of operations to tridiagonal form by a unitary congruence and, moreover,
the spectrum of $A$ is located on a straight line in the complex plane. In this
paper we present some generalizations of these properties for almost normal
matrices which satisfy certain quadratic matrix equations arising in the study
of structured eigenvalue problems for perturbed Hermitian and unitary matrices.
