Let $M$ be a $3$ $\times$ $3$ skew symmetric matrix with real entries.
Then I need to show that $M$ is diagonalizable over $\Bbb{C}$.
This has been my attempt.
The characteristic polynomial will be of degree $3$ and will have real coefficients. So if there are complex roots of this polynomial, they will be in pairs.
Since a skew symmetric matrix can only have eigen values either $0$ or purely imaginary, we can conclude that $0$ will definitely be an eigen value of $M$ since the complex ones are in pairs.
So there are two possibilities:-
- Eigen values of $M$ are $z_1, z_2$ and $0$ where $z_1$ and $z_2$ are complex numbers and conjugates of each other. In this case since eigen values are distinct, we can conclude that $M$ is diagonalizable over $\Bbb{C}$.
- Eigen values are $0, 0, 0$. From here, I can't conclude that $M$ is diagonalizable over $\Bbb{C}$.
I want help in second case.