Symmetric and Skew Symmetric Matrix

In science, commerce and even our everyday life it is often convenient to represent a set of numbers in rows and columns, called arrays. Suppose a company has three factories X, Y and Z which produces four commodities A, B, C and D. Further assume that X produces 4, 0, 7 and 6 units of A, B, C and D respectively; the corresponding units produced by Y and Z are 10, 3, 8, 5 and 8, 9, 2, 11 respectively. This information relating to production can be displayed by the following array:

Fig 1

Clearly, each row represents the number of units of a particular commodity produced by three factories and each column represents the number of units of different commodities produced in a particular factory. With this sense in advance the above array can be written as follows:

Fig 2

Such an array of numbers arranged in rows and columns is called a matrix. The above matric has 4 rows and three columns. This array is called a matrix of order 4 x 3. 

Definition: A matrix is a rectangular array of numbers which is arranged in rows and columns. If m.n numbers are arranged in a rectangular array of m rows and n columns, it is called a matrix of order m by n (written as mxn). 

Fig 3

The numbers a11, a12, a13, etc constituting a matrix are called elements or entries of the matrix. The matrix is the element in the ith row and jth column. The capital letters are used to denote matrices, 

Transpose of a Matrix

Let A be a matrix of order m x n; then the matrix of order n x m obtained by interchanging the rows and columns of A is called Transpose of the matrix A and is denoted by A’ or AT. For example, if


A = [fig 6] then A’= [fig 7]

Fig

Note that A is a matrix of order 3x2 and its transpose A’ is a matrix of order 2x3. 

Symmetric Matrix

A square matrix which is equal to its transpose is called a symmetric matrix. For example, a square matrix A = [aij] is symmetric if and only if aij= aji for all values of i and j, that is, if a12 = a21, a23 = a32, etc. Note that if A is a symmetric matrix then A’ = A where A’ is a transpose matrix of A.


Thus, A= [fig 4] is a symmetric matrix of order 3.

Fig

Note that the transpose of A = A’ = [fig5] = A.

Fig

How to Know if a Matrix is Symmetric

To know if the given matrix is symmetric or not, check the following conditions:

  1. It should be a square matrix.

  2. After transposing the matrix, it remains the same as that of the original matrix. 

Symmetric Matrix Properties

  1. The addition or subtraction of any two symmetric matrices will also be symmetric in nature.

  2. The product of two symmetric matrices [A and B] doesn’t always give a symmetric matrix [AB]. The result of the product is symmetric only if two individual matrices commute (AB=BA).

  3. The power on the symmetric matrix will also result in a symmetric matrix if the power n is integers.

  4. The inverse of a symmetric matrix is also asymmetric. 

Difference Between Symmetric and Skew-symmetric matrix


                    Symmetric Matrix

            Skew-Symmetric Matrix

Symmetric Matrix definition: Transpose of a matrix is always equal to the matrix itself. A^T=A


Skew-symmetric Matrix definition: Transpose of a matrix is always equal to the negative of the matrix itself.    A^T=-A


The main diagonal elements of a skew-symmetric matrix are not zero.

The main diagonal elements of a skew-symmetric matrix are zero.


Symmetric Matrix Example: 

 

Skew symmetric Matrix Example: 


Determinant of Matrix

A fixed number that defines a square matrix is called the determinant of a matrix. 

The process of finding the determinant of a symmetric matrix and the determinant of skew-symmetric is the same as that of a square matrix. 

Matrix inverse of a symmetric matrix

If A and B are two square matrices of the same order such that AB = BA = I, where I is the unit matrix of the same order as A. or B, then either B is called the inverse of A or A is called the inverse of B. The inverse of matrix A is denoted by A-1.

The inverse of a square matrix A exists if |A| is not equal to 0.

Let  A and B are two non-singular Matrices then,


i) A-1. A = A. A-1 = I 

ii) (A-1)-1=A

iii) (A-1)T = (AT)-1

iv) (AB)-1= B-1A-1

FAQ (Frequently Asked Questions)

Question 1: Is Every Symmetric Matrix Orthogonal?

Answer: No, not every symmetric matrix is an orthogonal matrix. We know that a matrix to be symmetric, its transpose must be equal to itself (A=AT) whereas, for a matrix to be orthogonal, its product with its orthogonal must be an Identity matrix (A. AT= I).


Thus, a symmetric matrix A is also orthogonal only if A2 = I 

Question 2: What is an Orthogonal Matrix?

Answer: A square matrix of order n is said to be orthogonal if A. AT= AT. A= I where AT is the transpose of A and I is the unit matrix of order n. 


If A is an orthogonal matrix then, 

A. AT= AT. A= I                ---------------------- (1)


Again, by the properties of determinants we have,

Det.AT = det.A and det.A. AT= Det.A Det.AT= (Det.A)2


From eq (1) we can say that,

det.A. AT= det I = 1 or (Det.A)2 = 1 or Det.A= 1


If det A = 1, then A is called a proper orthogonal matrix.

If det A = -1, then A is called an improper orthogonal matrix.