Orthogonal diagonalize this matrix

Question

how would one orthogonally diagonalize the matrix $A$ = $\begin{pmatrix}a&0&b\\ 0&a&0\\ b&0&a\end{pmatrix}$

where $b≠0$?

Answer 1

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.