Problem: If $A = \begin{pmatrix} -6 & 3 \\ 3 & -1\end{pmatrix}$ and $q(x)=x.Ax$ then $q(e_1)$ = ? and $q(e_2)$ = ?
Attempt: $$ x.Ax= \begin{pmatrix} x_1 \\ x_2\end{pmatrix}\begin{pmatrix} -6 & 3 \\ 3 & -1\end{pmatrix}\begin{pmatrix} x_1 \\ x_2\end{pmatrix} = -6x_1^2+6x_1x_2-1x_2^2 $$ x.Ax gets the same matrix as A.
I've been studying Quadratic Form and Matrix Norm. I know how to get eigenvalues and eigenvectors. But I don't understand what $q(e_1)$ and $q(e_2)$ are?