I've been solving the following problem on symmetrical bilinear forms:
Let $V$ a $\mathbb R$-vector space of dimension $n$ and $ f_1, f_2 \in V^\ast$ two non-zero linear functionals such that $f_1$ is not a scalar multiple of $f_2$. Define the application $f:V \times V \rightarrow \mathbb R$ given by:
$$f(u, v):= \frac{f_1(u)f_2(v)+f_2(u)f_1(v)}{2}.$$
Of course $f$ is a symmetrical bilinear form. What I don't know what to do is determine the signature of $f$. In fact, I believe that you need to compute the matrix of $f$ in some basis, and after that, find a diagonal matrix similar to the matrix of $f$ (it always exists). After that, count the number of negative and positive terms on the diagonal (which is always equal, by Sylvester's law of inertia).
However, I was a little confused on how to do this. My question would be: am I on the right path? Is there any simpler way to find the signature?
Also, I was stuck trying to follow after building the matrix of $f$ on some basis. If anyone could continue with any tips, i would be very grateful.
Hint
Take $\mathcal B =\{e_1, e_2, e_3, \dots,e_n \}$ as a basis of $V$ where $\operatorname{span}\{e_3, \dots, e_n\} =\ker f_1 \cap \ker f_2$ and $f_1(e_1) =f_2(e_2)=1$. This is possible considering the given hypothesis.
In $\mathcal B$, the matrix of $f$ is
$$\begin{pmatrix} 0 & 1/2 & 0 &\cdots &0\\ 1/2 & 0 & 0 &\cdots &0\\ 0 & 0 & 0 &\cdots &0\\ \vdots & \vdots & \vdots &\ddots &\vdots\\ 0 & \cdots & 0 &\cdots &0\\ \end{pmatrix}.$$ You’ll get that signature is $1,-1,0, \dots ,0$.