<s>[INST] Orthogonal diagonalize this matrix how would one orthogonally diagonalize the matrix $A$ = $\begin{pmatrix}a&0&b\\ 0&a&0\\ b&0&a\end{pmatrix}$
where $b≠0$?
 [/INST] This is a classic problem.  Read up on "Jacobi rotations"
Here is the short answer:
Focus on the matrix $ \left( \begin{array}{c c}
a & b\\
b & a 
\end{array}
\right)
$.
A Jacobi rotation is given by
$$ \left( \begin{array}{c c}
c & s\\
-s & c 
\end{array}
\right)
$$
where $ c = \cos( \theta ) $ and $ s = \sin( \theta ) $ 
such that, for this problem,
$$ \left( \begin{array}{c c}
c & s\\
-s & c 
\end{array}
\right)
\left( \begin{array}{c c}
a & b\\
b & a 
\end{array}
\right)
 \left( \begin{array}{c c}
c & s\\
-s & c 
\end{array}
\right)^T
$$
is diagonal.
You then take that, and use it as follows:
$$ \left( \begin{array}{c c}
c & 0 & s\\
0 & 1 & 0 \\
-s & 0 & c 
\end{array}
\right)
\left( \begin{array}{c c c }
a & 0 & b\\
0 & a & 0 \\
b & 0 & a 
\end{array}
\right)
\left( \begin{array}{c c}
c & 0 & s\\
0 & 1 & 0 \\
-s & 0 & c 
\end{array}
\right)^T
$$
The exact formula for $ c $ and $ s $ I don't quite remember.  
A rotation is an orthogonal matrix, so it what you need.
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