Symmetric and Skew Symmetric Matrix
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]
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Note that A is a matrix of order 3x2 and its transpose A’ is a matrix of order 2x3.
Symmetric Matrix
A square matrix that 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.
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Note that the transpose of A = A’ = [fig5] = A.
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How to Know If a Matrix is Symmetric
To know if the given matrix is symmetric or not, check the following conditions:
It should be a square matrix.
After transposing the matrix, it remains the same as that of the original matrix.
Symmetric Matrix Properties
The addition or subtraction of any two symmetric matrices will also be symmetric in nature.
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).
The power on the symmetric matrix will also result in a symmetric matrix if the power n is integers.
The inverse of a symmetric matrix is also asymmetric.
Difference Between Symmetric and Skew-Symmetric Matrix
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.
If A is nonsingular then, A-1 = \[\frac{adj(A)}{|A|}\]
Let A and B are two nonsingular 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
FAQs on Symmetric Matrix
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.
