For which $a$ and $b$ is this matrix diagonalizable? For which $a$ and $b$ is this matrix diagonalizable?

$$A=\begin{pmatrix} a &amp; 0 &amp; b \\ 0 &amp; b &amp; 0 \\ b &amp; 0 &amp; a  \end{pmatrix}$$

How to get those $a$ and $b$? I calculated eigenvalues and eigenvectors, but don't know what to do next?
That is a symmetric matrix: it is always diagonalizable over a field with characteristic$\;\neq2\;$. Not only that: it is orthogonally diagonalizable.
Matrix A has 3 eigenvalues a-b, a+b and b. In order for them to be distinguishable, following conditions should be met. $$b \ne 0$$ $$a \ne 0$$ $$a \ne 2b$$