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**1. The problem statement, all variables and given/known data**

Diagonalize the matrix $$ \mathbf {M} =

\begin{pmatrix}

1 & -\varphi /N\\

\varphi /N & 1\\

\end{pmatrix}

$$ to obtain the matrix $$ \mathbf{M^{'}= SMS^{-1} }$$

**2. Relevant equations**

First find the eigenvalues and eigenvectors of ##\mathbf{M}##, and then normalize the eigenvectors to get ##\mathbf{S^{-1}}##.

**3. The attempt at a solution**

After calculation, the eigenvalues are ## \lambda = 1 \pm \frac \varphi N##, and the corresponding eigenvectors (unnormalized) are ##\begin{pmatrix}

i \\ 1

\end{pmatrix}## and ##

\begin{pmatrix}

-i \\1

\end{pmatrix}

##.

Then I try to Schmidt them, but I failed to normalize them.

Could you help me normalize them?

Regards