This paper deals with the effect of generic but structured low rank perturbations on the Jordan structure and sign
characteristic of matrices that have structure in an indefinite inner product space.
The paper is a follow-up of earlier papers in which the effect of rank one perturbations
was considered. Several results that are in contrast to the case of unstructured low rank
perturbations of general matrices are presented here.
The inverse eigenvalue problem for $T$-alternating matrix polynomials over arbitrary
algebraically closed fields of characteristic different from two is considered.
The main result shows that the necessary conditions obtained in \cite{MacMMM10} for a matrix
polynomial to be the Smith form of a $T$-alternating matrix polynomial are under mild
conditions also sufficient to be the Smith form of a $T$-alternating matrix polynomial with
invertible leading coefficient which is additionally in anti-triangular form.. In particular, this result
implies that any $T$-alternating matrix polynomial with invertible leading coefficient is
equivalent to a $T$-alternating matrix polynomial in anti-triangular form that has the
same finite and infinite elementary divisors as the original matrix polynomial.
Finally, the inverse eigenvalue problem for $T$-palindromic matrix polynomials is
considered excluding the case that both $+1$ and $-1$ are eigenvalues.