In linear algebra, a symmetric matrix is identified as the square matrix that is equivalent to its transpose matrix. The transpose matrix of any assigned matrix say X, can be written as\(X^{T}\). A symmetric matrix Y can accordingly be represented as,\(Y=Y^{T}\). With all the various classes of matrices, symmetric matrices are one of the most prominent ones that are extensively used in machine learning.

With this article on the symmetric matrix, you will learn about the definition of the symmetric matrix with various properties, theorems, determinants, eigenvalues, the inverse of a symmetric matrix with solved examples and more.

A symmetric matrix in linear algebra is a square matrix that remains unchanged after taking its transpose. That proves that a matrix whose transpose is equivalent to the matrix itself is called a symmetric type of matrix.

A Matrix is depicted as an array of numbers(real or complex) that are arranged in rows(horizontal lines) and columns(vertical lines ). A rectangular representation of mn numbers (complex or real) in the form of m rows and n columns is named as a matrix of order m × n. Any matrix in which the number of rows is equivalent to the number of columns, say “n”, is termed as a square matrix of order n.

\(A=\begin{bmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\vdots&\vdots\\ a_{m1}&a_{m2}&\cdots&a_{mn}\end{bmatrix}\)

\(\text{i.e}\ A=[a_{ij}]_{m\times n\ }\text{is declared to be a square matrix of order n if}\ m=n.\)

This implies that a square matrix is a matrix where the number of columns is equal to the number of rows.

If m=n, the matrix is supposed to be a square matrix. If m ≠ n, the matrix is assumed to be a rectangular matrix. Here, m denotes the number of rows and n denotes the number of columns.

Learn about Transformation Matrix in the linked article!

A square matrix say, \(A=[a_{ij}]\) is recognised as a symmetric matrix if \(a_{ij}=a_{ji},\text{ for all}\ i,\ j.\) i.e a square matrix in mathematics that is identical to its transpose is recognised as a symmetric kind of matrix.

In other words if \(\text{If}\ B=[b_{ij}]_{n\times n}\) is the symmetric matrix, then

\(b_{ij}=b_{ji}\text{ for all}\ i\ \text{and}\ j\ \text{or}\ 1\le i\le n,\ \text{and}\ 1\le j\le n.\)

Transpose of Matrix

The matrix received from a given matrix Y by replacing its rows into columns/columns into rows is termed the transpose of matrix Y and is denoted by\(Y^T\text{or}\ Y’\).

Therefore if the order of the matrix Y before taking the transpose was m x n then the order of \(Y^T\) becomes n x m. Consider the below matrix example.

\(A=\begin{bmatrix}a_1&a_2&a_3\\ b_1&b_2&b_3\end{bmatrix}_{2\times3}\Rightarrow\ A^T=\begin{bmatrix}a_1&b_1\\ a_2&b_2\\ a_3&b_3\end{bmatrix}_{_{3\times2}}\)

Knowing about the square matrix and transpose of the matrix let’s proceed towards the symmetric matrices example.

Below is an example of a 2 × 2 symmetric matrix. A 2 × 2 symmetric type of matrix is an order 2 matrix, with 4 elements arranged in such a way that the transpose of the matrix is equal to the matrix itself.

\(P=\begin{bmatrix}\ 1&-3\\ -3&\ \ 0\end{bmatrix}\Rightarrow\ P^T=\begin{bmatrix}\ \ 1&-3\\ -3&\ \ 0\end{bmatrix}\)

Check out this article on identity matrix.

Similar to the 2 × 2 symmetric matrices we can have a 3×3 matrix as well as shown in the below diagram. Where a matrix of order 3 is taken having 9 elements arranged in such a way that the transpose of the matrix is equivalent to the matrix itself.

\(B=\begin{bmatrix}\ \ 1&4&-3\\ \ \ 4&1&\ 7\\ -3&7&\ 0\end{bmatrix}\Rightarrow B^T=\begin{bmatrix}\ \ 1&4&-3\\ \ \ 4&1&\ 7\\ -3&7&\ 0\end{bmatrix}\)

Well acknowledged with the definition and example of 2×2 and 3×3 matrices, let us learn about the properties relating to the article. Some of the important and frequently used symmetric matrices properties are listed below:

There are two important theorems associated with symmetric matrix and they are:

For any square matrix Q including real number elements:

\(Q+Q^T\) is a symmetric matrix, and \(Q-Q^T\) is a skew-symmetric matrix.

Any square matrix can be represented as the combination of a skew-symmetric matrix and a symmetric matrix.

\(Q=\left(\frac{Q+Q^T}{2}\right)+\left(\frac{Q-Q^T}{2}\right)\)

Similar to symmetric matrices there exist skew-symmetric or non-symmetric matrices under the various types of matrices. A matrix is said to be skew-symmetric if it is a square matrix and the transpose of a matrix is equivalent to the negative of that matrix; i.e If P is a symmetric matrix, then \(P=P^T\) and if P is a skew-symmetric matrix then \(-P=P^T\). This states that:\(p_{ij}=-p_{ji}\) for all the values of i and j. The “diagonal components” of a skew-symmetric matrix are equivalent to zero. Example of skew-symmetric matrix are given below:

\(Q=\begin{bmatrix}\ 0&\ \ 3\\ -3&\ \ 0\end{bmatrix}\)

\(Q=\begin{bmatrix}\ \ 0&\ \ 2&-3\\ -2&\ \ 0&\ 7\\ \ \ 3&-7&\ 0\end{bmatrix}\)

Determining the determinant of a symmetric matrix is similar to the determinant of the square matrix. Consider A be the symmetric matrix, and the determinant is indicated as \(\text{det A or}\ |A|\). Here, it relates to the determinant of matrix A. After some linear transform specified by the matrix, the determinant of the symmetric matrix is determined.

Learn about Cramer’s Rule in the linked article!

With the complete overview of symmetric matrices through definition, examples, properties, theorems and more, it’s time to practice some examples to implement the discussed concepts and understand them more distinctly.

Solved Example 1: If we are having a symmetric matrix say B, then verify that, \(B^T=B\).

Solution: Let us take a symmetric type of matrix as: \(B=\begin{bmatrix}2&3&6\\ 3&4&5\\ 6&5&9\end{bmatrix}\) Then:

\(B^T=\begin{bmatrix}2&3&6\\ 3&4&5\\ 6&5&9\end{bmatrix}\) Hence we can say that \(B^T=B\).

Also, read about various Matrix Operations here.

Solved Example 2: Check for the symmetricity of the given matrix: \(B=\begin{bmatrix}4&3\\ 2&1\end{bmatrix}\)

Solution: For a matrix to be symmetric \(B^T=B\) \(B^T=\begin{bmatrix}4&2\\ 3&1\end{bmatrix}\) Here we can see that \(B^T\ne B\) Therefore the matrix is nonsymmetric in nature.

Solved Example 3:The below matrix is symmetric or skew-symmetric in nature. \(P=\begin{bmatrix}0&-x\\ x&\ \ 0\end{bmatrix}\)

Solution: Given \(P=\begin{bmatrix}0&-x\\ x&\ \ 0\end{bmatrix}\)

Take the transpose of the matrix:

\(P^T=\begin{bmatrix}\ 0&x\\ -x&0\end{bmatrix}\)

\(P^T=\begin{bmatrix}\ 0&x\\ -x&0\end{bmatrix}=-P\) Hence the given matrix is skew-symmetric in nature.

We hope that the above article is helpful for your understanding and exam preparations. Stay tuned to the Testbook app for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.

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