# Subspaces of Symmetric, Skew-Symmetric Matrices

## Problem 143

Let $V$ be the vector space over $\R$ consisting of all $n\times n$ real matrices for some fixed integer $n$. Prove or disprove that the following subsets of $V$ are subspaces of $V$.

(a) The set $S$ consisting of all $n\times n$ symmetric matrices.

(b) The set $T$ consisting of all $n \times n$ skew-symmetric matrices.

(c) The set $U$ consisting of all $n\times n$ nonsingular matrices.

## Hint.

Recall that

• a matrix $A$ is symmetric if $A^{\trans}=A$.
• a matrix $A$ is skew-symmetric if $A^{\trans}=-A$.

## Proof.

To show that a subset $W$ of a vector space $V$ is a subspace, we need to check that

1. the zero vector in $V$ is in $W$
2. for any two vectors $u,v \in W$, we have $u+v \in W$
3. for any scalar $c$ and any vector $u \in W$, we have $cu \in W$.

### (a) The set $S$ consisting of all $n\times n$ symmetric matrices.

We will prove that $S$ is a subspace of $V$. The zero vector $O$ in $V$ is the $n \times n$ zero matrix and it is symmetric. Thus the zero vector $O\in S$ and the condition 1 is met.

To check the second condition, take any $A, B \in S$, that is, $A, B$ are symmetric matrices.
To show that $A+B \in S$, we need to check that the matrix $A+B$ is symmetric.

We have
\begin{align*}
(A+B)^{\trans}=A^{\trans}+B^{\trans}=A+B
\end{align*}
since $A, B$ are symmetric. Thus $A+B$ is also symmetric, and $A+B \in S$. Condition 2 is also satisfied.

Finally, to check condition 3, let $A \in S$ and let $r\in R$. We show that $rA \in S$, namely, we show that $rA$ is symmetric.
We have
\begin{align*}
(rA)^{\trans}=rA^{\trans}=rA
\end{align*}
since $A$ is symmetric. Thus $rA$ is symmetric and hence $rA \in S$.
Thus condition 3 is met.
By the subspace criteria, the subset $S$ is a subspace of the vector space $V$.

### (b) The set $T$ consisting of all $n \times n$ skew-symmetric matrices.

We will prove that $T$ is a subspace of $V$.
The zero vector $O$ in $V$ is the $n \times n$ matrix, and it is skew-symmetric because
$O^{\trans}=O=-O.$ Thus condition 1 is met.

For condition 2, take arbitrary elements $A, B \in T$. The matrices $A, B$ are skew-symmetric, namely, we have
$A^{\trans}=-A \text{ and } B^{\trans}=-B \tag{*}.$ We show that $A+B \in T$, or equivalently we show that the matrix $A+B$ is skew-symmetric.

We have
\begin{align*}
(A+B)^{\trans}=A^{\trans}+B^{\trans} \stackrel{*}{=} -A+(-B)=-(A+B).
\end{align*}
Therefore the matrix $A+B$ is skew-symmetric and condition 2 is met.

To prove the last condition, consider any $A \in T$ and $r \in \R$.
We show that $rA$ is skew-symmetric, and hence $rA \in T$.
Using the fact that $A$ is skew-symmetric ($A^{\trans}=-A$), we have
$(rA)^{\trans}=rA^{\trans}=r(-A)=-rA.$ Hence $rA$ is skew-symmetric and condition 3 is satisfied.

By the subspace criteria, the subset $T$ is a subspace of the vector space $V$.

### (c) The set $U$ consisting of all $n\times n$ nonsingular matrices.

We claim that $U$ is not a subspace of $V$.
As the zero vector of $V$ is the $n \times n$ matrix and the zero matrix is singular, the zero vector is not in $U$. Hence condition 1 is not met, and thus $U$ is not a subspace.

Another reason that $U$ is not a subspace is that the addition is not closed. For example,
if $A$ is a nonsingular matrix (say, $A$ is $n\times n$ identity matrix), then $-A$ is also nonsingular matrix but their addition $A+(-A)=O$ is nonsingular, hence it is not in $U$.

### More from my site

• Subspace of Skew-Symmetric Matrices and Its Dimension Let $V$ be the vector space of all $2\times 2$ matrices. Let $W$ be a subset of $V$ consisting of all $2\times 2$ skew-symmetric matrices. (Recall that a matrix $A$ is skew-symmetric if $A^{\trans}=-A$.) (a) Prove that the subset $W$ is a subspace of $V$. (b) Find the […]
• Quiz 8. Determine Subsets are Subspaces: Functions Taking Integer Values / Set of Skew-Symmetric Matrices (a) Let $C[-1,1]$ be the vector space over $\R$ of all real-valued continuous functions defined on the interval $[-1, 1]$. Consider the subset $F$ of $C[-1, 1]$ defined by $F=\{ f(x)\in C[-1, 1] \mid f(0) \text{ is an integer}\}.$ Prove or disprove that $F$ is a subspace of […]
• Rank and Nullity of a Matrix, Nullity of Transpose Let $A$ be an $m\times n$ matrix. The nullspace of $A$ is denoted by $\calN(A)$. The dimension of the nullspace of $A$ is called the nullity of $A$. Prove the followings. (a) $\calN(A)=\calN(A^{\trans}A)$. (b) $\rk(A)=\rk(A^{\trans}A)$.   Hint. For part (b), […]
• Diagonalizable by an Orthogonal Matrix Implies a Symmetric Matrix Let $A$ be an $n\times n$ matrix with real number entries. Show that if $A$ is diagonalizable by an orthogonal matrix, then $A$ is a symmetric matrix.   Proof. Suppose that the matrix $A$ is diagonalizable by an orthogonal matrix $Q$. The orthogonality of the […]
• Maximize the Dimension of the Null Space of $A-aI$ Let $A=\begin{bmatrix} 5 & 2 & -1 \\ 2 &2 &2 \\ -1 & 2 & 5 \end{bmatrix}.$ Pick your favorite number $a$. Find the dimension of the null space of the matrix $A-aI$, where $I$ is the $3\times 3$ identity matrix. Your score of this problem is equal to that […]
• Column Rank = Row Rank. (The Rank of a Matrix is the Same as the Rank of its Transpose) Let $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$.   Hint. Recall that the rank of a matrix $A$ is the dimension of the range of $A$. The range of $A$ is spanned by the column vectors of the matrix […]
• A Relation of Nonzero Row Vectors and Column Vectors Let $A$ be an $n\times n$ matrix. Suppose that $\mathbf{y}$ is a nonzero row vector such that $\mathbf{y}A=\mathbf{y}.$ (Here a row vector means a $1\times n$ matrix.) Prove that there is a nonzero column vector $\mathbf{x}$ such that $A\mathbf{x}=\mathbf{x}.$ (Here a […]
• Construction of a Symmetric Matrix whose Inverse Matrix is Itself Let $\mathbf{v}$ be a nonzero vector in $\R^n$. Then the dot product $\mathbf{v}\cdot \mathbf{v}=\mathbf{v}^{\trans}\mathbf{v}\neq 0$. Set $a:=\frac{2}{\mathbf{v}^{\trans}\mathbf{v}}$ and define the $n\times n$ matrix $A$ by $A=I-a\mathbf{v}\mathbf{v}^{\trans},$ where […]

#### You may also like...

This site uses Akismet to reduce spam. Learn how your comment data is processed.

##### Find a Value of a Linear Transformation From $\R^2$ to $\R^3$

Let $T:\R^2 \to \R^3$ be a linear transformation such that $T(\mathbf{e}_1)=\mathbf{u}_1$ and $T(\mathbf{e}_2)=\mathbf{u}_2$, where \$\mathbf{e}_1=\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}...

Close