Symmetric Matrix and Skew Symmetric Matrix

Symmetric Matrix is known that similar matrices have similar dimensions, thus only the square matrices can either be symmetric or skew-symmetric. In other words, it can be said that both a symmetric matrix and a skew-symmetric matrix are square matrices and the difference between a symmetric matrix and a skew-symmetric matrix is that symmetric matrix is always equivalent to its transpose while the skew-symmetric matrix is a matrix whose transpose is always equivalent to its negative. For example, If M is a symmetric matrix then M = MT and if M is a skew-symmetric matrix then M = - M\[^{T}\].

When a symmetric matrix and skew-symmetric matrix are summed up, the resultant matrix is always square.

Meaning of a Symmetric Matrix

A matrix cab only is stated as a symmetric matrix if its transpose is equivalent to the matrix itself. It should be remembered that only a square matrix is a symmetric matrix because in linear algebra similar matrices have similar dimensions.

Generally, the symmetric matrix is expressed as

M = M\[^{T}\]

Where M is any matrix and M\[^{T}\] is

transpose of that matrix.

If a(i,j) represents any

elements in an ith column

and jth rows,

then the symmetric matrix is expressed as 

aᵢⱼ = aⱼᵢ

Where every element of a asymmetric matrix is symmetric concerning the main diagonal whereas A square Matrix A can be defined as  skew-symmetric if aij = aji for all the values of i and j. So, we can also say that the matrix P is said to be the skew-symmetric if the transpose of matrix A is equal to the negative of Matrix A, In other words, A\[^{T}\] = −A.

What Is a Skew-Symmetric Matrix With an Example?

A square matrix A is defined as skew-symmetric if aᵢⱼ = aⱼᵢ for all the values of i and j. In other words, we can say that matrix P is said to be skew-symmetric if the transpose of matrix A is equal to the negative of Matrix A i.e (AT = −A). Also, it is important to note that all the elements present in the main diagonal of the skew-symmetric matrix are always zero. Let us understand this through a skew-symmetric matrix example.

Skew-Symmetric Matrix Example

The below skew-symmetric example helps you to clearly understand the concept of the skew matrix.

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In the above skew matrix symmetric example, we can see all the elements present in the main diagonal of matrices A are zero and also a₁₂ = -2 and  a₂₁ = -2 which implies that a₁₂ = a₂₁. This condition is valid for each value of I and j.

Properties of Skew-Symmetric Matrix

Some of the properties of skew-symmetric matrix examples are given below:

• When two skew-matrices are added, then the resultant matrix will always be a skew-matrix.

• The result of the scalar product of skew-symmetric matrices is always a skew-symmetric matrix.

• All the elements included in the main diagonal of the skew matrix are always equal to zero. Hence, the total of all the elements of the skew matrix in the main diagonal is zero.

• When both identity matrix and skew-symmetric matrix are added, the matrix obtained is invertible.

• The determinants of skew-symmetric matrices are always non-negative.

Solved Example

1. For the Given Below Matrix M, Verify That (M + M') Is a Symmetric Matrix.

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Solution:

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As, (M + M') = M + M'

Hence, (M + M') is a symmetric matrix.

2. Show That Matrix M Given Below is a Skew- Symmetric Matrix.

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Solution:

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∴, M = M’

Hence, M is a skew-symmetric matrix.