Eigenvalue decomposition of $2\times 2$ matrix

Question

There is a matrix $$A = \begin{bmatrix}3 & 1\\ 1 & 3 \end{bmatrix}$$ and the factorisation of $A = S*D*S^T$ needs to figured out. So far I have figured out that the eigensolutions are 4 and 2 and the corresponding vector spaces are $[1, 1]$ and $[-1, 1]$. From this point I am not sure how to proceed... So if someone could point me in the right direction, would appreciate it...

Answer 1

If $P$ is a matrix with eigenvectors of $A$ as its columns, then you can check that $AP = PD$. Right multiply by $P^{-1}$ and you should have a good idea what $S$ should be.

Be careful, one should normalize the columns of $P$ first so that $P^T = P^{-1}$.

Challenge: Can you see that $P^{-1}$ changes basis from the standard basis to basis of eigenvectors?