Diagonalizing a matrix with a symmetric matrix

Question

Ok.

The question was, find a real matrix $U$ with $U^{-1} = U^T$ Such that $A = UDU^T$ Where $D$ is diagonal matrix.

and $$A=\begin{bmatrix}1/2 & -3/2 \\ -3/2 & 1/2\end{bmatrix}$$

I get how to find any old $U$, that will diagonalize $A$. I have done that. But the problem I have is that the $U$ i found is not orthogonal (inverse!=transpose)

The $U$ i found is $$\begin{bmatrix}1 & -1 \\ 1 & 1\end{bmatrix}$$

How do I find a $U$ that is $U^{-1} = U^T$ ?

Answer 1

Almost there, just orthonormalise U and you are done.