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Symmetric and Skew Symmetric Matrices | Class 12 Maths

  • Last Updated : 26 Nov, 2020

The knowledge of matrices is necessary for various branches of mathematics. Matrices are one of the most powerful tools in mathematics. Now see one of the features of the matrix in this article. Here we discuss Symmetric and Skew Symmetric Matrices. According to matrices, only the square matrices can be symmetric or skew-symmetric form. Later in this article, we will discuss all things.

Symmetric Matrix

If for a matrix, the transposed form of that matrix is the same as the original matrix, then that matrix is said to be a Symmetric  Matrix. Let, a square matrix A of size n x n  is said to be symmetric if 

At = A

Where, [aij] = [aji], for 1 ≤ i ≤ n, and 1 ≤ j ≤ n. In this case [aij] is an element at position (i, j) which is ith row and jth column in matrix A, and [aji] is an element at position (j, i) which is jth row and ith column in matrix A. So it is the transposed form of matrix A.

Example



A =  \begin{pmatrix} 3 & 4 \\ 4 & 5 \end{pmatrix},
Transpose \ of \ A =  \begin{pmatrix} 3 & 4 \\ 4 & 5 \end{pmatrix}

So, this matrix is a Symmetric Matrix, because the transposed form of this matrix is itself the original matrix.

Skew-Symmetric Matrices

If for a matrix, the transposed form of that matrix is the same as the negative of the original matrix, then that matrix is said to be a Skew-Symmetric Matrix. Let, a square matrix A of size n x n  is  said to be skew-symmetric if

At = -A

Where, [aij] = [aji], for 1 ≤ i ≤ n, and 1 ≤ j ≤ n. In this case [aij] is an element at position (i, j) which is ith row and jth column in matrix A, and [aji] is an element at position (j, i) which is jth row and ith column in matrix A. 

Example

A =  \begin{pmatrix} 0 & 3 & -8 \\ -3 & 0 & 6 \\ 8 & -6 & 0 \end{pmatrix},
Transpose \ of \ A =  \begin{pmatrix} 0 & -3 & 8 \\ 3 & 0 & -6 \\ -8 & 6 & 0 \end{pmatrix}

Here, in the transposed form the matrix looks like the negative of the original matrix.



Properties of Symmetric and Skew-Symmetric Matrices

There are some rules that come from the concept of Symmetric and Skew-Symmetric Matrices,

1. If matrix A is a square matrix then (A + At) is always symmetric.

Prove: 

To find if a matrix symmetric or not, first, we have to find the transposed form of the given matrix 

So, let’s find the transpose of (A + At)

= (A + At)t

= At + (At)t

= At + A   [here, (At)t = A]

= (A + At)

So, this is the same as the given matrix, so it is symmetric.



2. If matrix A is a square matrix then (A – At) is always skew-symmetric.

Prove:

To find if a matrix skew-symmetric or not, first, we have to find the transposed form of the given matrix 

So, let’s find the transpose of (A – At)

= (A At)t

= At − (At)t

= At A  [here, (At)t = A]

= − (A At

So, this form is the negative of the given matrix, so it is skew-symmetric.

Expressing Matrix in the form of Symmetric and Skew-Symmetric Matrices

Every square matrix can be expressed uniquely as the sum of symmetric and skew-symmetric matrices.

Prove: 

Let A be a square matrix,

We can write, A = A/2 + A/2 

= \frac{1}{2}(A+A^t) + \frac{1}{2}(A-A^t)

Let, A = P + Q

where\quad P=\frac{1}{2}(A+A^t) ,\quad Q=\frac{1}{2}(A-A^t)

Now, find Pt and Qt

P^t =(\frac{1}{2}(A+A^t))^t \\ =\frac{1}{2}(A+A^t)^t \\ =\frac{1}{2}(A^t+(A^t)^t) \\ =\frac{1}{2}(A+A^t) \\ = P

and, 

Q^t =(\frac{1}{2}(A-A^t))^t \\ =\frac{1}{2}(A-A^t)^t \\ =\frac{1}{2}(A^t-(A^t)^t) \\ =\frac{1}{2}(A^t-A) \\ =-\frac{1}{2}(A-A^t) \\ = -Q



So, here P is symmetric and Q is skew-symmetric matrices and A is the sum of P and Q.

Example

Express matrix A as the sum of a symmetric and skew-symmetric matrix, Where 

A =  \begin{pmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{pmatrix}

Answer: 

First, find the transpose of A

A^t =  \begin{pmatrix} 0 & -a & -b \\ a & 0 & -c \\ b & c & 0 \end{pmatrix}

Now find (A + At) and (A – At)

(A+A^t) \\ =\begin{pmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{pmatrix} + \begin{pmatrix} 0 & -a & -b \\ a & 0 & -c \\ b & c & 0 \end{pmatrix} \\ =\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \\ \therefore \frac{1}{2}(A+A^t) = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}

Similarly:

(A-A^t) \\ =\begin{pmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{pmatrix} - \begin{pmatrix} 0 & -a & -b \\ a & 0 & -c \\ b & c & 0 \end{pmatrix} \\ =\begin{pmatrix} 0 & 2a & 2b \\ -2a & 0 & 2c \\ -2b & -2c & 0 \end{pmatrix} \\ \therefore \frac{1}{2}(A-A^t) \\ = \frac{1}{2}\begin{pmatrix} 0 & 2a & 2b \\ -2a & 0 & 2c \\ -2b & -2c & 0 \end{pmatrix} \\ = \begin{pmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{pmatrix}

Now, check the sum of (1/2)(A + At) and (1/2)(A – At) is the same as A or not,

\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}+\begin{pmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{pmatrix} \\ =\begin{pmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{pmatrix} \\ = A

So here A is expressed as the sum of the symmetric and skew-symmetric matrix.

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