Symmetric Matrix

A symmetric matrix is a square matrix that is equal to its transpose matrix. The transpose matrix of any given matrix A can be given as A^T. A symmetric matrix A, therefore, satisfies the condition, A = A^T. Among all the different kinds of matrices, symmetric matrices are one of the most important ones that are used widely in machine learning.

In this article, let's learn about symmetric matrices, their definitions, and properties with solved examples.

A symmetric matrix in linear algebra is a square matrix that remains unaltered when its transpose is calculated. That means, a matrix whose transpose is equal to the matrix itself, is called a symmetric matrix. It is mathematically defined as follows:

A square matrix B which of size n × n is considered to be symmetric if and only if B^T = B. Consider the given matrix B, that is, a square matrix that is equal to the transposed form of that matrix, called a symmetric matrix.

This can be represented as: If B = \(\left[\mathrm{b}_{\mathrm{ij}}\right]_{\mathrm{n} \times \mathrm{n}} \) is the symmetric matrix, then \(b_{ij}\) = \(b_{ji}\) for all i and j or 1 ≤ i ≤ n, and 1 ≤ j ≤ n. Here,

• n is any natural number.
• \(b_{ij}\) is an element at position (i, j) which is i^th row and j^th column in matrix B and
• \(b_{ji}\) is an element at position (j, i) which is j^th row and i^th column in matrix B.

Symmetric Matrix Examples

Let’s take an example of a matrix B,

Here, we can see that, B^T = B. For example, \(b_{12}\) = \(b_{21}\) = 3, and \(b_{13}\) = \(b_{31}\) = 6. Thus, B is a symmetric matrix. Given below are few more examples of symmetric matrices of different orders.

2 × 2 Symmetric Matrix Example: \(B = \left[\begin{array}{cc}
1 & -2 \\ \\
-2 & 0
\end{array}\right]\)

3 × 3 Symmetric Matrix Example: \(B = \left[\begin{array}{cc}
1 & 2 & -1 \\ 2 & 1 & 3 \\
-1 & 3 & 0
\end{array}\right]\)

4 × 4 Symmetric Matrix Example: \(B = \left[\begin{array}{cc}
1 & 2 & -1 & 5 \\ 2 & 1 & 3 & 0 \\
-1 & 3 & 0 & 4 \\ 5 & 0 & 4 & 2
\end{array}\right]\)

Here are some of the important properties of symmetric matrices.

• The sum and difference of two symmetric matrices give the resultant as a symmetric matrix.
• The property stated above is not always true for the product: Given the symmetric matrices A and B, 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 symmetric.
• The eigenvalues of a symmetric matrix are always real and positive.
• The determinant of a matrix and its transpose are same for a symmetric matrix.
• The adjoint of a symmetric matrix is symmetric.
• The inverse of symmetric matrix is symmetric.

There are two important theorems related to symmetric matrix. In this section, let's learn about these theorems along with their proofs.

Theorem 1: For any square matrix B with real number elements, B + B^T is a symmetric matrix, and B - B^T is a skew-symmetric matrix.

Proof:

Let A = B + B^T.

Taking a transpose, A^T = ( B + B^T )^T = B^T + ( B^T )^T = B^T + B = B + B^T = A

This implies B + B^T is a symmetric matrix.

Next, let C = B - B^T

C^T = ( B + ( - B^T ))^T = B^T + ( - B^T )^T = B^T - ( B^T )^T = B^T- B = - ( B - B^T ) = - C

This implies B − B^T is a skew-symmetric matrix.

Theorem 2: Any square matrix can be expressed as the sum of a skew-symmetric matrix and a symmetric matrix. To find the sum of a symmetric and skew-symmetric matrix, we use this formula:

Let B be a square matrix. Then,

B = (1/2) × (B + B^T) + (1/2 ) × (B - B^T). Here, B^T is the transpose of the square matrix B.

• If B + B^T is a symmetric matrix, then (1/2) × (B + B^T) is also a symmetric matrix
• If B - B^T is a skew-symmetric matrix, then (1/2 ) × (B - B^T) is also a skew-symmetric matrix

Thus, any square matrix can be expressed as the sum of a skew-symmetric matrix and a symmetric matrix.

Example: Express the following matrix as the sum of a symmetric and skew symmetric matrix:

\(B = \left[\begin{array}{cc}
1 & -1 & 4 \\ 2 & 1 & 3 \\
4 & 3 & 0
\end{array}\right]\)

Solution:

Since any matrix can be represented as a sum of a symmetric matrix and a skew symmetric matrix, we can therefore express matrix B as,
B = (1/2) × (B + B^T) + (1/2 ) × (B - B^T), where (1/2) × (B + B^T) is a symmetric matrix and (1/2) × (B - B^T) is a skew symmetric matrix.
⇒ (1/2) × (B + B^T) = (1/2) \( \left[\begin{array}{cc}
1 & -1 & 4 \\ 2 & 1 & 3 \\
4 & 3 & 0
\end{array}\right]\) + \(\left[\begin{array}{cc}
1 & 2 & 4 \\ -1 & 1 & 3 \\
4 & 3 & 0
\end{array}\right]\) = (1/2)\( \left[\begin{array}{cc}
2 & 1 & 8 \\ 1 & 2 & 6 \\
8 & 6 & 0
\end{array}\right]\) = \( \left[\begin{array}{cc}
1 & \frac{1}{2} & 4 \\ \frac{1}{2} & 1 & 3 \\
4 & 3 & 0
\end{array}\right]\)

Similarly, (1/2) × (B -B^T) = (1/2) \(\left[\begin{array}{cc}
1 & -1 & 4 \\ 2& 1 & 3 \\
4 & 3 & 0
\end{array}\right]\) - \( \left[\begin{array}{cc}
1 & 2& 4 \\ -1& 1 & 3 \\
4 & 3 & 0
\end{array}\right]\) = (1/2)\(\left[\begin{array}{cc}
0 & -3 & 0 \\ 3 & 0 & 0 \\
0 & 0 & 0
\end{array}\right]\) = \(\left[\begin{array}{cc}
0 & \frac{-3}{2} & 0 \\ \frac{3}{2} & 0 & 0 \\
0 & 0 & 0
\end{array}\right]\)

∴ Matrix B can be expressed as a sum of symmetric matrix and skew symmetric matrix as,
\( B = \left[\begin{array}{cc}
1 & -1 & 4 \\ 2 & 1 & 3 \\
4 & 3 & 0
\end{array}\right]\) = \( \left[\begin{array}{cc}
1 & \frac{1}{2} & 4 \\ \frac{1}{2} & 1 & 3 \\
4 & 3 & 0
\end{array}\right]\) + \( \left[\begin{array}{cc}
0 & \frac{-3}{2} & 0 \\ \frac{3}{2} & 0 & 0 \\
0 & 0 & 0
\end{array}\right]\)

Here, \(
\left[\begin{array}{cc}
1 & \frac{1}{2} & 4 \\ \frac{1}{2} & 1 & 3 \\
4 & 3 & 0
\end{array}\right]\) is a symmetric matrix and \( \left[\begin{array}{cc}
0 & \frac{-3}{2} & 0 \\ \frac{3}{2} & 0 & 0 \\
0 & 0 & 0
\end{array}\right]\) is a skew symmetric matrix.

The symmetric and skew-symmetric matrices share a close relationship with each other. There is one major difference between symmetric matrix and skew-symmetric matrix. The differences between symmetric and skew-symmetric matrices are explained in the below-given table:

Important Notes on Symmetric Matrices:

Here is a list of a few points that should be remembered while studying symmetric matrices.

• A square matrix that is equal to the transposed form of itself is called a symmetric matrix.
• Since all off-diagonal elements of a square diagonal matrix are zero, every square diagonal matrix is symmetric.
• The sum of two symmetric matrices gives a symmetric matrix as result.

☛ Related Topics:

Check out the following pages related to the symmetric matrix.

• Matrix Calculator
• Matrix formula
• Diagonal Matrix Calculator
• Transpose Matrix Calculator

What is Symmetric Matrix Definition?

A square matrix that is equal to the transpose of that matrix is called a symmetric matrix. An example of a symmetric matrix is given below,

\(A=\left[\begin{array}{ll}
2 & 7\\ \\
7 & 8
\end{array}\right]\)

How Do You Know if a Matrix is a Symmetric Matrix?

To check if any given matrix is a symmetric matrix, we find the transpose of the given matrix and compare the resultant transpose matrix with the existing matrix. If both the matrices are equal, then the given matrix is a symmetric matrix.

What are the Properties of Symmetric Matrix?

These are the important properties of symmetric matrices that distinguish them from other types of matrices. These properties are given as,

• The sum/difference matrix of two symmetric matrices is a symmetric matrix.
• The above property doesn't need to be true for the multiplication of matrices: for two symmetric matrices A and B, product matrix AB is symmetric if and only if they are commutative, i.e., if AB = BA.
• If A is symmetric, Aⁿ is also symmetric, for any integer n.
• If the inverse of a matrix A exists, it will be symmetric if and only if the square matrix A is symmetric.

What is the Example of a Symmetric Matrix?

For any symmetric matrix, A, A = A^T. The matrix A given below is an example of a symmetric matrix.

\(A=\left[\begin{array}{ll}
1 & 2\\ \\
2 & 3
\end{array}\right]\)

What is the Sum of Two Symmetric Matrices?

As per the properties of the symmetric matrices, the sum of any two symmetric matrices will always be a symmetric matrix. Consider two symmetric matrices A and B, then A + B = C, C will also be a symmetric matrix.

What is the Sum of a Skew Symmetric and Symmetric Matrix?

As per the properties of the symmetric and skew-symmetric matrices, the sum of any symmetric and skew-symmetric matrix is always a square matrix. If B is a square matrix then, B = (1/2) × (B + B^T) + (1/2 ) × (B - B^T). Here, B^T is the transpose of the square matrix B, B + B^T is a symmetric matrix, and B - B^T is a skew-symmetric matrix.

How Do You Find if a Matrix is a Symmetric Matrix?

These are the steps to find a symmetric matrix:

• Step 1: First, check if it's a square matrix, as only square matrices can be considered symmetric matrices.
• Step 2: Find the transpose of the given matrix.
• Step 3: If the transpose of the matrix is equal to the matrix itself, then it is a symmetric matrix.

What is the Difference Between Symmetric and Skew Symmetric Matrices?

The difference between symmetric and skew-symmetric matrix has been explained in the below:

• A symmetric matrix is a square matrix B which is of size n × n, is considered to be symmetric if and only if B^T = B, whereas a skew-symmetric matrix is a square matrix B which is of size n × n, is considered to be symmetric if and only if B^T = -B.
• For symmetric matrix, \(b_{ij}\) = \(b_{ji}\) and for skew-symmetric matrix \(b_{ij}\) = - \(b_{ji}\).