On Fiedler's characterization of tridiagonal matrices over arbitrary fields

Fiedler proved in [Linear Algebra Appl. 2 (1969) 191-197] that the set of real n-by-n symmetric matrices A such that rank(A + D) ⩾ n − 1 for every real diagonal matrix D is the set of matrices PTPT where P is a permutation matrix and T an irreducible tridiagonal matrix. We show that this result remains valid for arbitrary fields with some exceptions for 5-by-5 matrices over Z3.