The congruence of a matrix with its transpose (Q1781915)

For a square matrix \(A\) over any field \(F\) it was proved recently that \(A\) is congruent to its transpose \(A^t\), in the sense that \(P^{t}AP=A^t\), for some nonsingular matrix \(P\). For bilinear forms it means that any bilinear form \(b(x,y)\) on a finite-dimensional vector space is equivalent to its transposed form \(b(y,x)\). All proofs of the congruence result are based mostly on the modern theory of general bilinear forms developed by \textit{C. Riehm} [J. Algebra 31, 45--66 (1974; Zbl 0283.15016)]. Here, a short review of the Riehm's theory is presented and a new proof of the above mentioned congruence result is derived. Then, this result is applied to the theory of rings to show that certain rings \(R\) are isomorphic to its opposite \(R^o\) or equivalently that \(R\) possess an antiautomorphism.