I have a math question that I got stuck on and would like to ask you about:
I have a symmetric bilinear form on $\mathbb{Q}^4$ which is described w.r.t the standard basis.
$$g(v,w)=\mathbf{v}^\top Aw$$ where $$A= \begin{pmatrix} 1 & 2 & 3 & 4\\ 2 & 3 & 4 & 5\\ 3 & 4 & 5 & 6\\ 4 & 5 & 6 & 7\end{pmatrix} $$
a) I need to find a basis on whch g is diagonal. - I see that the rank is 2, so two eigenvalues must be $0$, but for the basis, I do need to find all 4 eigenvectors right? Or is there perhaps an easier way to see a basis.
b)I need to find one ( or a product of ) matrix (matrices) B (or B=$B_1\cdot B_2 \cdot\cdot$) such that $\mathbf{B}^\top AB$= diagonal
for this one I would suggest $B$ to be orthogonal, and I would use the orthogonal eigenvector from a) and normalise them, right?
c)Consider the same g on $\mathbb{R}^4$, I have to give a basis for which g is diagonal and has only $1,-1,0$ on the diagonal.
I dont know to connect this with my given information.
Any help would be greatly appreciated! Thanks in advance
