Symmetric Matrix - Definition, Properties, Theorems, Examples

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 AT. A symmetric matrix A, therefore, satisfies the condition, A = AT. 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.

1.	What is Symmetric Matrix?
2.	Properties of Symmetric Matrix
3.	Symmetric Matrices Theorems
4.	Difference between Skew Symmetric and Symmetric Matrix
5.	FAQs on Symmetric Matrix

What is Symmetric Matrix?

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 BT = 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 ith row and jth column in matrix B and
• \(b_{ji}\) is an element at position (j, i) which is jth row and ith column in matrix B.

Symmetric Matrix Examples

Let’s take an example of a matrix B,

Here, we can see that, BT = 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: \(B = \left[\begin{array}{cc} 1 & 2 & -1 & 5 \\ 2 & 1 & 3 & 0 \\ -1 & 3 & 0 & 4 \\ 5 & 0 & 4 & 2 \end{array}\right]\)

Properties of Symmetric Matrix

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.

Symmetric Matrices Theorems

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 + BT is a symmetric matrix, and B - BT is a skew-symmetric matrix.

Proof:

Let A = B + BT.

Taking a transpose, AT = ( B + BT )T = BT + ( BT )T = BT + B = B + BT = A

This implies B + BT is a symmetric matrix.

Next, let C = B - BT

CT = ( B + ( - BT ))T = BT + ( - BT )T = BT - ( BT )T = BT- B = - ( B - BT ) = - C

This implies B − BT 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 + BT) + (1/2 ) × (B - BT). Here, BT is the transpose of the square matrix B.

• If B + BT is a symmetric matrix, then (1/2) × (B + BT) is also a symmetric matrix
• If B - BT is a skew-symmetric matrix, then (1/2 ) × (B - BT) 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 + BT) + (1/2 ) × (B - BT), where (1/2) × (B + BT) is a symmetric matrix and (1/2) × (B - BT) is a skew symmetric matrix. ⇒ (1/2) × (B + BT) = (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 -BT) = (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.

Difference Between Skew Symmetric and 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:

Symmetric Matrix	Skew Symmetric Matrix
A square matrix B which is of size n × n, is considered to be symmetric if and only if BT = B.	A square matrix B which is of size n × n, is considered to be symmetric if and only if BT = -B.
Here, \(b_{ij}\) = \(b_{ji}\).	Here, \(b_{ij}\) = - \(b_{ji}\).
There is nothing specific about the determinant of symmetric matrix.	The determinant of a skew-symmetric matrix of odd order is 0.
The eigenvalues of the symmetric matrix are real.	The eigenvalues of the skew-symmetric matrix are purely imaginary.
The elements of the principal diagonal may be any elements.	The elements of the principal diagonal are always zeros.

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