Congruences of a square matrix and its transpose Original Research Article

It is known that any square matrix A over any field is congruent to its transpose: AT=STAS for some nonsingular S; moreover, S can be chosen such that S2=I, that is, S can be chosen to be involutory. We show that A and AT are *congruent over any field image of characteristic not two with involution image (the involution can be the identity): image for some nonsingular S; moreover, S can be chosen such that image, that is, S can be chosen to be coninvolutory. The short and simple proof is based on Sergeichukʹs canonical form for *congruence [Math. USSR, Izvestiya 31 (3) (1988) 481]. It follows that any matrix A over image can be represented as A=EB, in which E is coninvolutory and B is symmetric.