Introduction

Symmetric Matrices are those matrices where the transpose of a matrix is equal to the matrix itself. Suppose our original matrix is A, then the transpose of the matrix will be AT.

Thus, we can define symmetric matrices as:

A = AT

All symmetric matrices are square matrices, but not all square matrices are symmetric. We can identify symmetry by seeing the diagonals of a matrix. If there is symmetry along the diagonals of an original matrix and its transpose, it is symmetric.  

Another way to represent symmetric matrices is as follows:

A-1 AT = I

Here I is an identity matrix. An identity matrix is given as:

I =

A matrix that is not symmetric is known as an Asymmetric matrix. Although, do remember not to confuse an Asymmetric matrix with an Antisymmetric matrix, as both things are different. 

This study material notes on symmetric matrices contain more details on this topic, so keep reading to learn more. 

What is a Matrix?

A matrix is an array of numbers arranged in rows (horizontal lines) and columns (vertical lines). For matrix A, we can name the rows as m, while the columns are n. It gives rise to a matrix of order m × n.

So if they are 2 rows and 1 column, then the order of the matrix will be 2 × 1. For example:

The most common type of matrix is the 2 × 2. These are also known as the square matrices. For instance: 

Note:

• If the number of columns and rows is equal then it is a square matrix
• If the number of columns and rows is not equal then it is a rectangular matrix

How to Find the Transpose of a Matrix?

The transpose of a matrix can be found by interchanging the rows of the matrix with the columns. If a matrix is of order m × n, then the order of the transpose matrix will be n × m. 

Assume that we have a matrix of order:

Now, if we interchange the rows and columns, we get:

To illustrate:

Matrix A =

The transpose of this matrix will be: 

AT =

Symmetric Matrices

If the transpose of a matrix is the same as the original matrix, then it forms a symmetric matrix (Plural matrices)

Example of a symmetric matrix:

Suppose we have a 2 × 2 square matrix given as A. Now,

A =

Then, the transpose of A will be given by AT.  

AT = 

Since AT = A, it is a symmetric matrix. 

Only square matrices can be symmetric matrices. It means that only matrices with equal rows and columns can be symmetric matrices. 

3 × 3 Square Matrix Examples:

Suppose we have a 3 × 3 square matrix A. Then, 

A =

The transpose of the above matrix will be given by interchanging the first row with the first column, the second row with the second column, and the third row with the third column. 

AT =

Since A = AT, it is a symmetric matrix. 

Properties of Symmetric Matrices

1. The addition of two symmetric matrices results in a symmetric matrix.
2. The difference between two symmetric matrices results in a symmetric matrix.
3. If two matrices, A and B, are symmetric and follow the commutative property, which says the product of AB = the product of BA, then the resulting matrix will be symmetric. 
4. If matrix A is symmetric for an integer n, then An will also be symmetric.
5. A-1 will be symmetric only if A is symmetric. 

Skew Symmetric Matrix

A skew-symmetric matrix is one whose transpose is the negative of itself. The following formula represents it:

Matrix A = – AT

The symmetric skew matrix is always a square matrix. For example:

A =

AT= 

Thus, A = – AT

Symmetric Matrix Theorems

Two important theorems are associated with the symmetric matrix. These are:

Theorem 1:  

For square matrices, the sum of its matrix and the transpose of the said matrix will be symmetric, while the difference will be skew-symmetric. That is, for a matrix A and transpose AT, the sum A + AT is symmetrical while A – AT is skew-symmetric.

Theorem 2: 

Any square matrix can be expressed as the sum of a symmetric matrix and skew-symmetric matrix.

Difference between Skew Symmetric Matrix and Symmetric Matrix

1. The transpose matrix of a symmetric matrix is equal to the original matrix. While in a skew-symmetric matrix, the transpose matrix is negative of the original matrix. 
2. The sum of the diagonals in a skew-symmetric matrix is equal to zero. This fact is not true in the case of a symmetric matrix. 

Eigenvalues of Symmetric and Skew Symmetric Matrices

The symmetric matrices consist of real eigenvalues.While the eigenvalues of a skew-symmetric matrix are zero or complete imaginary or nonreal.

Conclusion

In this study material notes on symmetric matrices, we learned that symmetric matrices are those matrices whose transpose is equal to the original matrix. We also learned how to find the transpose of a matrix. The transpose of a matrix can easily be found by interchanging the rows and columns of a matrix. 

Another type of symmetric matrix is Skew Symmetric Matrix. It is the negative version of the transpose of a matrix. 

Symmetric matrices find their uses in machine learning and data science. Hence, it is one of the most indispensable topics in matrices. These symmetric matrices study material aims to aid the understanding of this topic easier.