On a certain canonical form (Q1585647)

The problem of reducing a real rectangular matrix \(A\) of even dimensions \(2m \times 2n\) to a canonical form by transformations of the form \(PAQ\), where \(P\) and \(Q\) are square nondegenerate formally complex matrices of order \(2m\) and \(2n\), respectively, consisting of \(2 \times 2\) blocks of the type \[ \left(\begin{matrix} x & y\\ -y & x \end{matrix}\right), \] where \(x, y\) are real numbers, is studied. The previous well-known result obtained by \textit{V. Dlab} and \textit{C. M. Ringel} [Linear Algebra Appl. 17, 107-124 (1977; Zbl 0382.15008)] is extended in the following way. By certain permutations of rows and columns of the matrix \(A\) it is obtained the equivalent problem of reducing a new real matrix \(\bar A\), divided into two horizontal and two vertical strips, by admissible transformations of its rows and columns.