When does matrices represent the same bilinear form

Question

Here is the task I was recently asked to solve:

At what value of the paramter $a$ matrices

\begin{equation*} A = \begin{pmatrix} 1 & 4-a-a^2 \\ 2 & -1\\ \end{pmatrix} \end{equation*}

and \begin{equation*} B = \begin{pmatrix} -a-1 & 3 \\ 3 & -5\\ \end{pmatrix} \end{equation*}

can be the matrices of a same bilinear form $V \times V \rightarrow \mathbb{R}$ in different bases?

I know that if the matrices $A$ and $B$ represent the same bilinear form then $$\exists \space S: A=S^TBS (1)$$

But what am I supposed to do next? Is it possible to solve equation $(1)$ for $S$?

I found the same post here: Find the values of parameter a so that matrices A and B - But there is no reasonable answer(why does $A$ has to be symmetric?)

Answer 1

You can notice that B is symmetric while A isn’t for every a. When you impose the condition on symmetry (you find the 2 values of a for wich A is symmetric) you can use spectral theory to determine wether or not they represent the same bilinear form. Obviously I consider that both are expressed in canonical bases.

Answer 2

Given two bases for the same vector space, there exist a matrix, S, that transforms the vectors in one basis to the basis in the other basis. Then a matrix, A, in one basis is the same as the basis $SAS^{-1}$ in the other.