So am trying to find signature of a bilinear form that satisfies $ f\mathrm (e_i\mathrm , e_j \mathrm)$ for all $i, j$ where $\mathrm e_i$ are the standard unit vectors.
So the only way I can think how to do this is to form a matrix $\mathrm A$ that is an $n \times n$ real symmetric matrix with $1$ in every entry and try find the number of positive, negative, and zero eigenvalues.
But now I'm stuck after constructing this matrix as its $n \times n$ and the usual method of $det\mathrm(A\mathrm - cI\mathrm)=0$ isn't possible, or is it? I don't know how to find all the eigenvalues for $n \times n$ matrix as have only computed for $2 \times 2$, $3 \times 3$, etc. No idea how to do it for $n \times n$ even if I do know all the entries.
As A is real symmetric matrix I know there should be $n$ real eigenvalues not necessarily distinct. Using definition of eigenvalues I think $c=0$ and $c=n$ may be two and $n$ could have multiplicity $n-1$ but that's not exactly proving it. Can anyone tell me what methods are available for computing eigenvalues for $n \times n$ if you know all the entries?
Or if there is a different way I could approach this problem? Just to clarify, I'm only a first year undergrad so have not come across a lot of the theory and possible methods. Oh and this is homework but I'm looking for advice, not just entire solution, as I want to know how I can do this when faced with similar situation for arbitrarily sized matrices.
Thanks for any hints or advice.