Congruence verification for involutive matrices

A finite computational process using only arithmetic operations is called a rational algorithm. Presently, no rational algorithm for checking the congruence of arbitrary complex matrices $A$ and $B$ is known. The situation may be different if both $A$ and $B$ belong to a special matrix class. For instance, there exist rational algorithms for the cases where both matrices are Hermitian, unitary, or accretive. In this publication, we propose a rational algorithm for checking the congruence of involutive matrices $A$ and $B$.