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

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with "problem I have is that the U i found is not symmetric" you mean that "problem I have is that the U i found is not orthogonal" or not? Beeing symmetric ($A=A^T$) is something different then beeing orthogonal ($A^{-1}=A^T$) – born Oct 25 '12 at 10:39
OH. I think that solves my question. I just need to orthonormalise U... right? – straykiwi Oct 25 '12 at 10:45
@straykiwi: Yes. – wj32 Oct 25 '12 at 10:45
So if you know how to do it now, you could post an answer to your question. Then, later, you can accept it. – Gerry Myerson Oct 25 '12 at 12:11
up vote 0 down vote accepted

Almost there, just orthonormalise U and you are done.

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