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 $U^{-1} = U^T$ ?