## 7 Problems on Skew-Symmetric Matrices

## Problem 564

Let $A$ and $B$ be $n\times n$ skew-symmetric matrices. Namely $A^{\trans}=-A$ and $B^{\trans}=-B$.

**(a)** Prove that $A+B$ is skew-symmetric.

**(b)** Prove that $cA$ is skew-symmetric for any scalar $c$.

**(c)** Let $P$ be an $m\times n$ matrix. Prove that $P^{\trans}AP$ is skew-symmetric.

**(d)** Suppose that $A$ is real skew-symmetric. Prove that $iA$ is an Hermitian matrix.

**(e)** Prove that if $AB=-BA$, then $AB$ is a skew-symmetric matrix.

**(f)** Let $\mathbf{v}$ be an $n$-dimensional column vecotor. Prove that $\mathbf{v}^{\trans}A\mathbf{v}=0$.

**(g)** Suppose that $A$ is a real skew-symmetric matrix and $A^2\mathbf{v}=\mathbf{0}$ for some vector $\mathbf{v}\in \R^n$. Then prove that $A\mathbf{v}=\mathbf{0}$.