What is the determinant of every skew symmetric matrix?

There are many different types of matrices in linear algebra which are differentiated based on their elements, order, and certain set of conditions. Matrices are referred to the plural form of a matrix.

What are the different types of matrices?

Below is a list of the most commonly used types of matrices in linear algebra:

·     Row matrix

·     Column matrix

·     Singleton matrix

·     Rectangular matrix

·     Square matrix

·     Identify matrices

·     Matrix of ones

·     Zero matrix

·     Diagonal matrix

Symmetric matrix

In linear algebra, a symmetric matrix is described as the square matrix which is equal to its transpose matrix. The transpose matrix of any given matrix A can be written as At. Therefore, a symmetric matrix A can be written as A = At. Out of all the different kinds of matrices, symmetric matrices are most commonly used in machine learning.

What is symmetric matrix?

In linear algebra, a symmetric matrix is a square matrix which remains unchanged when its transpose is calculated. In other words, a matrix whose transpose is equal to the matrix itself is known as a symmetric matrix. 

Symmetric matrix definition

A square matrix having size n * n is said to be symmetric if and only if Bt = B. 

Symmetric matrix example:

Given below is an example of a matrix B,

It can be seen that, B^T = B, b₁₂ = b₂₁ = 3. and b₁₃ = b₃₁ = 6.

So, B is a symmetric matrix.

Below are few more examples of symmetric matrices of different orders.

Properties of symmetric matrix

A matrix can be symmetric only when it is a square matrix- which means that it should have the same number of rows and columns. Some of the important properties of symmetric matrix are:

·  The sum and the difference of two symmetric matrices states the result as a symmetric matrix.

· The property which is mentioned above does not hold true for the product: Suppose the symmetric matrices and B are given, then AB is symmetric if and only if A and B follow commutative property of multiplication, i.e., if AB = BA.

·  For integer n, if A is symmetric, ➩ Aⁿ is also symmetric.

·  When A^-1 is there, it will be symmetric if and only if A is symmetric.

 

Symmetric matrix theorems

There are two important theorems of symmetric matrix.

Theorem 1:

This theorem states that any square matrix B with real number elements, B + Bt is a symmetric matrix, and B – Bt is a skew-symmetric matrix.

This proves that B – B t is a skew-symmetric matrix.

Theorem 2: 

Any square matrix can be said to be the sum of a skew symmetric matrix and a symmetric matrix. We can find out the sum of a symmetric and skew symmetric matrix by using the following formula:

Let us suppose that B is a square matrix. Then,

Therefore, any square matrix can be defined as the sum of a symmetric and skew symmetric matrix:

Example: Let us express the following matrix as the sum of a symmetric and skew symmetric matrix:

Solution:

We know that any matrix can be expressed as a sum of a symmetric matrix and a skew symmetric matrix. Therefore, matrix B can be expressed as: 

Therefore, matrix B can be represented as a sum of symmetric matrix and skew symmetric matrix as given below:

Difference between skew symmetric and symmetric matrix

The symmetric and skew-symmetric matrices have a close bonding with each other. But there is only one major difference between the two which is given below in the table:

Important notes on symmetric matrix

Below are given few points which should be kept in mind while studying symmetric matrices.

·    A square matrix which is equal to the transposed form of itself is called a symmetric matrix.

·  All off-diagonal elements of a square diagonal matrix are zero. Therefore, every square diagonal matrix is symmetric.

·    The result of the sum of two symmetric matrices is a symmetric matrix.

Determinant of Skew-Symmetric Matrix is equal to zero if its order is odd

If there is any skew-symmetric matrix with odd order, then its determinant is equal to zero. 

Therefore, the determinant of skew symmetric matrix is equal to zero.

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