For which values of $a$ matrix $A$ is diagonalizable? $$A = \pmatrix{0&i\\i&a}$$
in the case that it is not diagonalizable determine a base of Jordan
Attempt: The minimal polynomial already factored in is: $p_A(x)=(x-(\frac{a-\sqrt{a^2-4}}{2}))(x-(\frac{a+\sqrt{a^2-4}}{2}))$.
If they are the two distinct roots then $A$ is diagonalizable because its minimum polynomial is equal to the characteristic one and we see that it has different linear factors. On the other hand the roots of the polynomial coincide when $a= \pm 2$. It is verified that the minimum polynomial is not $(x \pm 1)$, because it does not annul $A$ (when $a= \pm 2$). In this case $A$ is not diagonalizable and we will find the base of Jordan.Is it correct here?