## Eigenvalues of Real Skew-Symmetric Matrix are Zero or Purely Imaginary and the Rank is Even

## Problem 269

Let $A$ be a real skew-symmetric matrix, that is, $A^{\trans}=-A$.

Then prove the following statements.

**(a)** Each eigenvalue of the real skew-symmetric matrix $A$ is either $0$ or a purely imaginary number.

**(b)** The rank of $A$ is even.