Given matrix $\begin{pmatrix} 1 & 3 \\ 3 & 1 \end{pmatrix}$, find a matrix $P$ such that $P^{-1}AP$ is diagonal and an orthogonal matrix $Q$ such that $Q^{-1}AQ$ is diagonal.
For the first part, I found the eigenvalues to be $\lambda = -2, 4$ and the associated eigenvectors to be $\begin{bmatrix} -1 \\ 1 \end{bmatrix}$ and $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$, so these are my basis for $\beta$ and $P = \begin{pmatrix} -1 & 1 \\ 1 & 1 \end{pmatrix}$. For $Q$, I was told that I need to use the Gram-Schmidt process on the basis vectors, but I'm unsure if that's correct and if that's the right approach. Can someone clarify?