Space of invariant bilinear forms (Q1787180)

Let \(V\) be a vector space of dimension \(n\) over a field \(\mathbb{F}\). Then the set of bilinear forms on \(V\) is a vector space of dimension \(n^2\) over \(\mathbb{F}\). Assume \(\operatorname{char} \mathbb{F} \neq 2\). If \(T\) is an invertible linear map from \(V\) onto \(V\) then the set \(\mathcal{B}\) of \(T\)-invariant bilinear forms is a subspace of this space of forms. The authors determine the dimension of the subspace \(\mathcal{B}\) under various assumptions for the map \(T\), e.g., if \(T\) is invertible, cyclic, and self reciprocal, then the dimension of \(\mathcal{B}\) is \(2n\); if \(T\) is indecomposable, self-reciprocal on \(V\), then the dimension of \(\mathcal{B}\) is \(n\).