I am having a hard time trying to prove the following:
Let $M$ be the space of all $2\times2$ complex matrices,
which is skew-hermitian ( i.e. $\bar{X}^t = -X$ ).
We consider M as a vector space over R,
For any $A\in M$,
define the operator $\text{ad}_A : M\to M $ by $\text{ad}_A (X) = AX – XA$.
Show that $\text{ad}_A$ is diagonalizable.
Let $$A=\left[\begin{array}{cc}i & 1\\-1 & i\end{array}\right].$$ How should I compute the eigenvalues of the operator $\text{ad}_A$ ?
(Maybe we should show there is a basis of $M$ consisting of eigenvectors of $\text{ad}_A$ ?)
Thanks.