On Matrices Having $$J_{k}(1)\oplus J_{l}(1)$$ as Their Cosquare

Rational techniques for verifying the congruence of complex matrices are discussed. An algorithm is said to be rational if it is finite and uses arithmetical operations only. An important part in verifying the congruence of nonsingular matrices play their cosquares. The verification gets complicated if there are eigenvalues of modulus 1 in the spectrum of cosquares; this is especially true if such eigenvalues are defective. In this direction, the most advanced result is the rational algorithm for matrices $$A$$ and $$B$$ whose cosquare is the direct sum $$J_{m}(1)\oplus J_{m}(1)$$ . Here, this algorithm is extended to the case where the cosquare is the direct sum of two Jordan blocks of distinct orders. This extension is heavily dependent on additional facts concerning the solutions to the matrix equation $$X-J_{m}^{\top}(1)XJ_{m}(1)=0$$ , which are found in the present paper.