Special congruences of symmetric and Hermitian matrices and their invariants

Let $V_n$ be the arithmetic space of dimension $n$, and let the inner product be introduced in $V_n$ using a symmetric or a skew-symmetric involution $M$. In the resulting indefinite metric space, one can define the classes of special matrices playing the parts of symmetric, skew-symmetric, and orthogonal operators. We say that such matrices are $M$-symmetric, $M$-skew-symmetric, and $M$-orthogonal, respectively. The invariants of $M$-orthogonal congruences performed with $M$-symmetric and $M$-skew-symmetric matrices are indicated. A Hermitian counterpart of these constructions is also discussed.