Symmetric Matrix & Skew Symmetric Matrix

To understand if a matrix is a symmetric matrix, it is very important to know about transpose of a matrix and how to find it. If we interchange rows and columns of an m×n  matrix to get an n × m   matrix, the new matrix is called the transpose of the given matrix. There are two possibilities for the number of rows (m) and columns (n) of a given matrix:

  • If m = n , the matrix is square
  • If m ≠ n , the matrix is rectangular

Symmetric matrix and Skew Symmetric Matrix

For the second case, the transpose of a matrix can never be equal to it. This is because, for equality, the order of the matrices should be the same. Hence, the only case where the transpose of a matrix can be equal to it is when the matrix is square. But this is only the first condition. Even if the matrix is square, its transpose may or may not be equal to it. For example:

If \( A =
\begin{bmatrix}
1& 2\cr
3 & 4
\end{bmatrix} \)
, then \( A’ =
\begin{bmatrix}
1& 3\cr
2 & 4
\end{bmatrix} \)
. Here, we can see that A ≠ A’ .

Let us take another example.

\( B =
\begin{bmatrix}
1& 2&17\cr
2 & 5&-11 \cr
17&-11&9
\end{bmatrix} \)

If we take the transpose of this matrix, we will get:

\( B’ =
\begin{bmatrix}
1& 2&17\cr
2 & 5&-11 \cr
17&-11&9
\end{bmatrix} \)

We see that B = B’. Whenever this happens for any matrix, that is whenever transpose of a matrix is equal to it, the matrix is known as a symmetric matrix. But how can we find whether a matrix is symmetric or not without finding its transpose? We know that:

If A = \( [a_{ij}]_{m×n}\) then A’ = \( [a_{ij}]_{n×m}\) ( for all the values of i and j )

So, if for a matrix A,\(a_{ij}\) = \(a_{ji}\) (for all the values of  and ) and m = n , then its transpose is equal to itself. A symmetric matrix will hence always be square. Some examples of symmetric matrices are:

\( P =
\begin{bmatrix}
15& 1\cr
1 & -3
\end{bmatrix} \)

\( Q =
\begin{bmatrix}
-101 & 12 & 57\cr
12 & 1001 & 23 \cr
57 & 23 & -10001
\end{bmatrix} \)

Skew Symmetric Matrix:

A matrix can be skew symmetric only if it is square. If the transpose of a matrix is equal to the negative of itself, the matrix is said to be skew symmetric. This means that for a matrix  to be skew symmetric,

A’=-A

Also, for the matrix,\(a_{ji}\) = – \(a_{ij}\)(for all the values of i and j). The diagonal elements of a skew symmetric matrix are equal to zero. This can be proved in following way:

The diagonal elements are characterized by the general formula,

\( a_{ij} \) , where i = j

If i = j, then \( a_{ij}\) = \( a_{ii}\) = \( a_{jj}\)

If A is skew symmetric, then

aji = – aji

⇒ aii = – aii

⇒ 2.aii = 0

⇒ aii = 0

So, aij = 0 , when i = j  (for all the values of i and j )

Some examples of skew symmetric matrices are:

\( P =
\begin{bmatrix}
0 & -5\cr
5 & 0
\end{bmatrix} \)

\( Q =
\begin{bmatrix}
0 & 2&-7\cr
-2 & 0&3 \cr
7 & -3 &0
\end{bmatrix} \)

Every square matrix can be expressed in the form of sum of a symmetric and a skew symmetric matrix in an unique way. Learn various concepts in maths & science by visting our site BYJU’S.


Practise This Question

If A and B are two square matrices which are skew symmetric then (AB)T equals to