On Fiedler's characterization of tridiagonal matrices over arbitrary fields (Q1779413)

An application of the general theorem, treated as Fiedler's characterization of tridiagonal matrices [\textit{M. Fiedler}, Linear Algebra Appl. 2, 191--197 (1969; Zbl 0194.06105)], to the case of tridiagonal \(5\times 5\)-matrices is considered. This theorem (Fiedler's property) is formulated in the paper as follows: ``Let \(A\) be an \(n\)-by-\(n\) real diagonal matrix. Then \(\text{rank}(A+ D)\geq n- 1\), for every \(n\)-by-\(n\) real diagonal matrix \(D\), if and only if \(A\) is permutational similar to an irreducible tridiagonal matrix''. As a result, some peculiarities, extensions and limitations, related to the basic relation \(A\to P^TTP\) in the case under consideration, where \(P\) is a permutation matrix and \(T\) an irreducible tridiagonal matrix, are presented. They are obtained and argued by making direct use of the explicit expressions for a wide set of the above mentioned \(5\times 5\)-matrices and the corresponding submatrices.