Orthogonal similarity and pairs of quadratic forms (Q1906765)

Let \(L/K\) be a finite separable field extension and let \(W\) be a \(K\)-space with an endomorphism \(S\) such that \(L = K[S]\). For any \(L\)-bilinear form \(C\) on \(W\) the trace map \(\text{Tr} = \text{tr}_{L/K}\) provides us with a \(K\)-bilinear form \(B = \text{Tr} \circ C\) such that \(B(Sx, y) = B(x,Sy)\) for \(x,y \in W\). In the paper the author examines the inverse process, that is, he shows how to get (up to isometry) the form \(C\) when the form \(\text{Tr} \circ C\) is known. As a result he gets that the two known ways of classifying rational symmetric matrices under orthogonal similarity, first presented by \textit{S. Friedland} [Linear Algebra Appl. 192, 109-114 (1993; Zbl 0787.15011)] and the second by the author [Invent. Math. 37, 157-164 (1976; Zbl 0337.10015)], are in fact the same.