Symmetric and Skew Symmetric Matrix


What is a Symmetric Matrix?

  • A square matrix which is equal to its transpose is known as a symmetric matrix.

  • Only square matrices are symmetric because only equal matrices have equal dimensions.

  • A matrix A with nn dimensions is said to be skew symmetric if and only if 

aij = aji for all i, j such that 1≤n, j≤n.

  • Suppose A is a matrix, then if the transpose of matrix A = AT is equal then it is a symmetric matrix.

  • Symmetric matrix example,

 

                 A     =

            The transpose of A (AT) = 

       Since, A= AT matrix A is a symmetric matrix.


NOTE: If any diagonal matrix is equal to the transpose of the matrix, such matrices are automatically symmetric.


Before We Move Further, Let us Know About Some Important Terms!

  • Square matrix is a matrix where the number of columns is equal to the number of rows.

If m=n, the matrix is a square matrix

If m ≠ n, the matrix is a rectangular matrix


                  Here, m = The number of rows

                     n = The number of columns


Transpose of a Matrix (AT)

We find the transpose of a matrix by interchanging the rows and columns of the original matrix.

Suppose the original matrix is denoted by n×m, the transpose of the matrix will be m×n.

Let us take an example, 

If A= , then let us calculate the transpose of the matrix A.

Here, the first row becomes the first column and the second row becomes the second column.

AT=

Here we see that A AT.

If A = , then let us calculate the transpose of the matrix A.

Here, the first row becomes the first column, the second row becomes the second column and the third row becomes the third column.

AT =

Here, we see that A = AT


How to Check Whether a Matrix is Symmetric or Not?

Step 1- Find the transpose of the matrix.

Step 2- Check if the transpose of the matrix is equal to the original matrix.

Step 3- If the transpose matrix and the original matrix are equal , then the matrix is symmetric.


Skew Symmetric Matrix Definition –

  • A square matrix is said to be skew symmetric if the transpose of the matrix equals its negative.

  • A matrix A with nn dimensions is said to be skew symmetric if and only if 

aij = -aji for all i, j such that 1≤n, j≤n.

  • Suppose A is a matrix, then if the transpose of matrix A, AT =- A is equal then it is a skew symmetric matrix.

For Example: 

Example 1

              A = 

            - A=    = AT

                Since, AT=-A matrix A is a skew symmetric matrix


Example 2

             P =   

            -P =    = PT

              Since, PT=-P matrix A is a skew symmetric matrix.

Conditions for Symmetric and Skew Symmetric Matrix 

SYMMETRIC MATRIX(A)

AT=A

aji=(aij)

SKEW SYMMETRIC MATRIX (A)

AT=(-A)

aji=(-aij)


                                   Here, i = Row entry

                                             j = Column entry


How to Check Whether a Matrix is Skew Symmetric or Not?

Step 1 - First find the transpose of the original given matrix.

Step 2 – Then find the negative of the original matrix.

Step 3 – If the negative of the matrix obtained in Step2 is equal to the transpose of    

              the matrix then the matrix is said to be skew symmetric.

PROPERTY :

Any matrix A can be written as a sum of /symmetric matrix and a skew symmetric matrix.

A = A+A T2+A-A T2


Questions to Solve

Question 1

 Check whether the given matrices are symmetric or not.

  • M =   


  • P =      


Solution

We will first find the transpose of matrix M,

MT =

Since the transpose of M is not equal to matrix M, therefore it is not a symmetric matrix. 

We will first find the transpose of matrix P,

PT =

Since the transpose of P is not equal to matrix P, therefore it is not a symmetric matrix. 


Question 2 

Is the given matrix A, a skew symmetric matrix. Give reason for your answer.

 

Solution

First, we will find the transpose of the matrix A,

Now we will find the negative of the matrix A.

Since, the negative of the matrix A is equal to the transpose of the matrix A. 

Therefore, A is a skew symmetric matrix.


Question 3

Show that the given matrix is a symmetric matrix.

                      

Solution

To check whether the given matrix A is a symmetric matrix,

We need to find the transpose of the given matrix A,

Since the original matrix A is equal to the transpose matrix, therefore the given matrix A is a symmetric matrix.


Question 4

Check whether the given matrix B is a symmetric matrix or a skew symmetric matrix.

                

Solution

Let’s check whether the given matrix is symmetric or not.

We need to find the transpose of the given matrix B,

Since the original matrix B is not equal to the transpose matrix (BT≠B), therefore the given matrix B is not a symmetric matrix.

Let’s check whether the given matrix is skew symmetric or not.

Since we have already found the transpose,

We will find the negative of the original matrix.

Since, the negative of the matrix B is equal to the transpose of the matrix B. 

Therefore, B is a skew symmetric matrix.

FAQ (Frequently Asked Questions)

1. What is a Symmetric matrix? Give an example.

A symmetric matrix is a matrix whose transpose is equal to the matrix itself. Example of a symmetric matrix:

The transpose of A (AT) =

 

2.  Is the zero-matrix symmetric?


Yes, the zero matrix is a symmetric matrix.

3. What are symmetric and skew symmetric matrices?

A symmetric matrix is a matrix whose transpose is equal to the matrix itself whereas a skew symmetric matrix is a matrix whose transpose is equal to the negative of itself.

Symmetric matrix example:

The transpose of A, 

 Skew-Symmetric matrix example:

4. Can we diagonalize a Symmetric Matrix?

Yes, a Symmetric matrix can always be diagonalized.