# Symmetric and Skew Symmetric Matrices

Here you will learn what are symmetric and skew symmetric matrices with examples.

Let’s begin –

## Symmetric and Skew Symmetric Matrices

#### Symmetric Matrix

A square matrix A = $$[a_{ij}]$$ is called a symmetric matrix, if

$$A^T$$ = A or $$a_{ij}$$ = $$a_{ji}$$ for all i, j.

for example, the matrix A = $$\begin{bmatrix} 3 & -1 & 1 \\ -1 & 2 & 5 \\ 1 & 5 & -2 \end{bmatrix}$$ is symmetric, because

$$a_{12}$$ = -1 = $$a_{21}$$, $$a_{13}$$ = 1 = $$a_{31}$$, $$a_{23}$$ = 5 = $$a_{32}$$ i.e. $$a_{ij}$$ = $$a_{ji}$$ for all i, j.

It follows from the definition of a symmetric matrix that A is symmetric, iff

$$a_{ij}$$ = $$a_{ji}$$   for all i, j.

$$\iff$$  $$A_{ij}$$ = $$(A^T)_{ij}$$  for all i, j

$$\iff$$  A = $$A^T$$

Thus, a square matrix A is a symmetric matrix iff $$A^T$$ = A.

Matrices A = $$\begin{bmatrix} a & h & g \\ h & b & f \\ g & f & c \end{bmatrix}$$, B = $$\begin{bmatrix} 2 + i & 1 & 3 \\ 1 & 2 & 3 + 2i \\ 3 & 3 + 2i & 4 \end{bmatrix}$$ are symmetric matrices because $$A^T$$ = A and $$B^T$$ = B.

#### Skew-Symmetric Matrix

A square matrix A = $$[a_{ij}]$$ is called a skew-symmetric matrix, if

$$A^T$$ = -A or $$a_{ij}$$ = -$$a_{ji}$$ for all i, j.

for example, the matrix A = $$\begin{bmatrix} 0 & 2 & -3 \\ -2 & 0 & 5 \\ 3 & -5 & 0 \end{bmatrix}$$ is skew symmetric, because

$$a_{12}$$ = 2, $$a_{21}$$ = -2 $$\implies$$ $$a_{12}$$ = -$$a_{21}$$ $$a_{13}$$ = -3, $$a_{31}$$ = 3 $$\implies$$ $$a_{13}$$ = -$$a_{31}$$

and,  $$a_{23}$$ = 5,  $$a_{32}$$ = – 5 $$\implies$$ $$a_{23}$$ = -$$a_{32}$$

It follows from the definition of a skew symmetric matrix that A is skew symmetric, iff

$$a_{ij}$$ = -$$a_{ji}$$   for all i, j.

$$\iff$$  $$A_{ij}$$ = -$$(A^T)_{ij}$$  for all i, j

$$\iff$$  A = -$$A^T$$

Thus, a square matrix A is a skew symmetric matrix iff $$A^T$$ = -A.

Matrices A = $$\begin{bmatrix} 0 & 2i & 3 \\ -2i & 0 & 4 \\ -3 & -4 & 0 \end{bmatrix}$$, B = $$\begin{bmatrix} 0 & -3 & 5 \\ 3 & 0 & 2 \\ -5 & -2 & 0 \end{bmatrix}$$ are skew symmetric matrices because $$A^T$$ = -A and $$B^T$$ = -B.