# 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 $A\mathbf{v}=\mathbf{0}$.

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Contents

- Problem 564
- Proof.
- (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 $A\mathbf{v}=\mathbf{0}$.

## Proof.

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

We have

\begin{align*}

(A+B)^{\trans}=A^{\trans}+B^{\trans}=(-A)+(-B)=-(A+B).

\end{align*}

Hence $A+B$ is skew-symmetric.

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

We compute

\begin{align*}

(cA)^{\trans}=cA^{\trans}=c(-A)=-cA.

\end{align*}

Thus, $cA$ is skew-symmetric.

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

Using the properties of transpose, we have

\begin{align*}

(P^{\trans}AP)^{\trans}&=P^{\trans}A^{\trans}(P^{\trans})^{\trans}=P^{\trans}A^{\trans}P\\

&=P^{\trans}(-A)P=-(P^{\trans}AP).

\end{align*}

This implies that $P^{\trans}AP$ is skew-symmetric.

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

Note that since $A$ is real, we have $\bar{A}=A$.

Then we have

\begin{align*}

(\overline{iA})^{\trans}=(\bar{i}\bar{A})^{\trans}=(-iA)^{\trans}=(-i)A^{\trans}=(-i)(-A)=iA.

\end{align*}

It follows that $iA$ is Hermitian.

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

We calculate

\begin{align*}

(AB)^{\trans}&=B^{\trans}A^{\trans}=(-B)(-A)\\

&=BA=-AB,

\end{align*}

where the last step follows from the assumption $AB=-BA$.

This proves that $AB$ is skew-symmetric.

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

Observe that $\mathbf{v}^{\trans}A\mathbf{v}$ is a $1\times 1$ matrix, or just a number.

So we have

\begin{align*}

\mathbf{v}^{\trans}A\mathbf{v}&=(\mathbf{v}^{\trans}A\mathbf{v})^{\trans}=\mathbf{v}^{\trans}A^{\trans}(\mathbf{v}^{\trans})^{\trans}\\

&=\mathbf{v}^{\trans}A^{\trans}\mathbf{v}=\mathbf{v}^{\trans}(-A)\mathbf{v}=-(\mathbf{v}^{\trans}A\mathbf{v}).

\end{align*}

This yields that $2\mathbf{v}^{\trans}A\mathbf{v}=0$, and hence $\mathbf{v}^{\trans}A\mathbf{v}=0$.

### (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 $A\mathbf{v}=\mathbf{0}$.

Let us compute the length of the vector $A\mathbf{v}$.

We have

\begin{align*}

\|A\mathbf{v}\|&=(A\mathbf{v})^{\trans}(A\mathbf{v})=\mathbf{v}^{\trans}A^{\trans}A\mathbf{v}\\

&=\mathbf{v}^{\trans}(-A)A\mathbf{v}=-\mathbf{v}^{\trans}A^2\mathbf{v}\\

&=-\mathbf{v}\mathbf{0} &&\text{by assumption}\\

&=0.

\end{align*}

Since the length $\|A\mathbf{v}\|=0$, we conclude that $A\mathbf{v}=\mathbf{0}$.

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