(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

First off, this is not for a course, I'm reviewing material. This also *should* be straightforward! I think I'm forgetting something simple, so if someone could point it out to me, I would be able to sleep easy tonight :)

OK! the question:

Given a matrix [tex]T = \begin{pmatrix}

cosx & -sinx\\

sinx & cosx\\

\end{pmatrix}$[/tex]

we want to find the inverse of [tex]{\bf S}^{-1}={\bf S}[/tex] and then take [tex]STS^{-1}[/tex].

2. Relevant equations

So first I find the eigenvalues which are cosx +/- isinx

Next I calculated the eigenvectors and got a(1) = (1,-i) [column vector]

and a(2) = (1,i) [column vector]

If you normalize the two eigenvectors you get a constant 1/sqrt[2] for both.

That gives me

[tex]S^{-1} = \frac{1}{\sqrt{2}}\begin{pmatrix}

1 & 1\\

-i & i\\

\end{pmatrix}$[/tex]

and when I solve I get [tex]S = \frac{1}{\sqrt{2}}\begin{pmatrix}

1/2 & i/2\\

1/2 & -i/2\\

\end{pmatrix}$[/tex]

3. The attempt at a solution

If you look above, you should see that I did most everything correctly (I believe, let me know if I made an error)! However, clearly [tex]STS^{-1}[/tex] should give me back a matrix with my eigenvalues on the diagonal. However, I get an extra coefficient in front of the matrix which should cancel out. Where is the mistake? Also, along those lines, I read somewhere that [tex]S=(S^{-1})^{\dag}[/tex] is this true? I was always under the impression that S is simply the inverse and you do not need to take the adjoint?

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# Homework Help: Diagonalizing Martrix (S) question

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