Symmetric Matrix and Skew Symmetric Matrix: Properties of Symmetric Matrix

Get unlimited access to the best preparation resource for KVPY : fully solved questions with step-by-step explanation- practice your way to success.

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 , the matrix is square

  • If , the matrix is rectangular

Symmetric Matrix

Symmetric Matrix

Loading Image •••

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 then Here, we can see that

Let us take another example.

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

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 then (for all the values of i and j)

So, if for a matrix (for all the values of i and j) and then its transpose is equal to itself. A symmetric matrix will hence always be square. Some examples of symmetric matrices are:

Properties of Symmetric Matrix

  • Addition and difference of two symmetric matrices results in symmetric matrix.

  • If A and B are two symmetric matrices and they follow the commutative property, i.e. , then the product of A and B is symmetric.

  • If matrix A is symmetric then and is also symmetric, where n is an integer.

  • If A is a symmetric matrix then A-1 is also symmetric.

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,

Also, for the matrix,(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,

, where

If then

If A is skew symmetric, then

⇒

⇒

⇒

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

Some examples of skew symmetric matrices are:

Properties of Skew Symmetric Matrix

  • When we add two skew-symmetric matrices then the resultant matrix is also skew-symmetric.

  • Scalar product of skew-symmetric matrix is also a skew-symmetric matrix.

  • The diagonal of skew symmetric matrix consists of zero elements and therefore the sum of elements in the main diagonals is equal to zero.

  • When identity matrix is added to skew symmetric matrix then the resultant matrix is invertible.

  • The determinant of skew symmetric matrix is non-negative

Every square matrix can be expressed in the form of sum of a symmetric and a skew symmetric matrix

Frequently Asked Questions

How Do You Know if a Matrix is Symmetric?

To know if a matrix is symmetric, find the transpose of that matrix. If the transpose of that matrix is equal to itself, it is a symmetric matrix.

Give an Example of a Matrix Which is Symmetric but Not Invertible

A zero (square) matrix is one such matrix which is clearly symmetric but not invertible.

Is Symmetric Matrix Diagonalizable?

Yes, a symmetric matrix is always diagonalizable.

Example 1: Construct the matrix , where . State whether A is symmetric or skew – symmetric.

Solution:

From the given question, we come to know that we have to construct a matrix with 3 rows and 3 columns.

So, the matrix A with order is

Whether it is symmetric or skew symmetric matrix.

Hence, it is skew symmetric matrix.