On a commutator result of Taussky and Zassenhaus

In a recent paper [1], Taussky and Zassenhaus showed that A is nonderogatory if and only if any nonsingular X in the null space of T is symmetric. In this note we investigate the structure of the null space of both T and T for arbitrary A. Enlarge the field F to include λ4, i = 1, , p, the distinct eigenvalues of A, and let (x — Xt) tj, j = 1, , nt, etl > > eiUi, i = 1, , p be the distinct elementary divisors of A where (x — \)u appears with multiplicity rυ. Set m4 = Σ"=i^«βw, the algebraic multiplicity of λ̂ . Let f]{T) denote the null space of T, σ(T) denote the subspace of symmetric matrices in ?](T), and y(T) denote the subspace of skew-symmetric matrices in η(T). We show that