Symmetric Matrix in Discrete mathematics

In case of discrete mathematics, we can define a symmetric matrix as a square matrix that is similar to its transpose matrix. We can call a matrix as square matrix if it contains the same number of rows and columns. If there is a matrix A, then the transpose of this matrix will be indicated by the symbol A^T. If a matrix satisfies the condition A = A^T, then that matrix will be a symmetric matrix. We have several types of matrices, but the most important type of matrix, which is mostly used is the symmetric matrix. In this section, we are going to learn about the symmetric matrix, its properties, theorems of a symmetric matrix, the difference between a symmetric matrix and a skew-symmetric matrix, examples of a symmetric matrix and many more.

What is symmetric matrix

If there is a square matrix, then we can call it a symmetric matrix if we calculate the transpose of this matrix and it remains unchanged. That means in case of a symmetric matrix, the original matrix and its transpose matrix will be the same.

Definition of symmetric matrix

Suppose there is a square matrix B which contains the size n∗n. This matrix will be considered as symmetric iff B^T = B. If this given square matrix B and the transpose of this matrix are similar to each other, then we can call this matrix as the symmetric matrix. The symmetric matrix can be represented in the following way:

If there is a symmetric matrix B = [bij]_n∗n, then bij = bji for all i and j or 1 ≤ i ≤ n. Here

• n is used to show any natural number
• bij is used to show an element at position (i, j). In the given matrix B, i is used to show the row, and j is used to showing the column.
• bji is used to indicate an element at position (j, i). In the given matrix B, j is used to show the row, and i is used to showing the column.

Transpose of a matrix

A matrix will be known as the transpose matrix if we interchange the elements of rows or columns of the original matrix. We can indicate the transpose of a matrix with the help of symbol T. If there is a matrix which is indicated by n∗m, then its transpose matrix will be indicated by m∗n. Now we will understand the concept of transpose by an example, which is described as follows:

Example 1: Suppose we have a 2∗2 matrix A. Here we have to determine the transpose of matrix A. The elements of matrix A are described as follows:

Solution: For the transpose matrix, we will change the first row into the first column, and the second row will be changed into the second column in the following way:

Here A ≠ A^T. So this matrix is not a symmetric matrix.

Example 2: In this example, we have a 3∗3 matrix A. We have to determine the transpose of matrix A. The elements of matrix A are described as follows:

Solution: To perform the transpose of this matrix, we will change the first row into the first column and the second row will be changed into the second column, and the third row will be changed into the third column in the following way:

Here A = A^T. Hence it is a symmetric matrix.

Steps to determine the symmetric matrix

There are some steps which we should be followed to determine whether the matrix is symmetric matrix or not. These steps are described as follows:

Step 1: In this step, we will determine the transpose of a matrix.

Step 2: In this step, we will check whether the original matrix and the transpose of this matrix are similar to each other or not.

Step 3: The matrix will be known as the symmetric matrix if the original matrix and the transpose matrix are similar to each other.

Now we will understand this concept with the help of an example, which is described as follows:

Suppose we have a matrix A where

The transpose of this matrix is described as follows:

The original matrix A and its transpose matrix A^T are similar to each other. Hence A is a symmetric matrix.

Symmetric matrix Examples

Now we will understand the symmetric matrix with the help of an example. Suppose we have a 3∗3 square matrix, B where

The transpose of matrix B is described as follows:

In this example, we have B^T = B. For example, b12 = b21 = 3 and b13 = b31 = 6. Thus, we can say that B is a symmetric matrix. Now we will show some more examples of symmetric matrices, but with different orders.

Example 1: In this example, we will show a symmetric matrix with an order 2∗2, i.e.,

In this matrix, we can see that we have a square matrix that contains 4 elements. These elements are arranged in the form of 2 rows and 2 columns.

Example 2: In this example, we will show a symmetric matrix with an order 3∗3, i.e.,

In this matrix, we can see that we have a square matrix that contains 9 elements. These elements are arranged in the form of 3 rows and 3 columns.

Example 3: In this example, we will show a symmetric matrix with an order 4∗4, i.e.,

In this matrix, we can see that we have a square matrix that contains 16 elements. These elements are arranged in the form of 4 rows and 4 columns.

Properties of Symmetric matrix

There are various properties of a symmetric matrix, and some of them are described as follows:

• If there are two symmetric matrices and we perform the addition or subtraction on these matrices, then we will always get the symmetric matrix as a resultant matrix.
• In case of multiplication operation, we will not always get a symmetric matrix as a result. Suppose we have two symmetric matrices, A and B, and we perform the multiplication operation on these matrices. In this case, the resultant matrix AB will be symmetric iff these matrices follow the commutative properties of multiplication, i.e., AB = BA.
• If there is a symmetric A, then Aⁿ will also be symmetric, where n is used to indicate the integer n.
• If there exists an inverse matrix A^-1, then it will be symmetric if and only if the matrix A is symmetric.

Theorems of Symmetric matrix

In case of a symmetric matrix, we have two important theorems. Here we will learn about both the theorems as well as their proofs.

Theorem 1: If there is a square matrix B, which contains the real number elements, in this case, B + B^T will be a symmetric matrix and B - B^T will be a skew-symmetric matrix.

Proof:

Here we will assume that

A = B + B^T

Now we will take the transpose of assumed matrix A and get the following:

A^T = (B + B^T)^T= B^T+ (B^T)^T= B^T+ B = B + B^T= A

This shows that B + B^T is a symmetric matrix.

After that, we will assume that

C = B - B^T

Now we will take the transpose of assumed matrix C and get the following:

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

This shows that B - B^T is a skew-symmetric matrix.

Theorem 2: If there is a square matrix, then we can write this matrix as the addition of a symmetric matrix and skew-symmetric matrix. With the help of following formula, we can determine the addition of a skew-symmetric matrix and a symmetric matrix.

Suppose there is a square matrix B. Then,

B = (1/2) × (B + B^T) + (1/2) × (B - B^T). Here transpose of matrix B is shown with the help of symbol B^T.

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

Hence if there is a square matrix, then we can write this matrix as the addition of a skew-symmetric matrix and symmetric matrix.

Example: In this example, we have a matrix B, and we have to express it in the form of addition of a symmetric matrix and skew-symmetric matrix. The elements of matrix B are described as follows:

Solution: As we have learned that we can write this matrix as the addition of a skew-symmetric matrix and a symmetric matrix. So we can also express matrix B in the following way:

B = (1/2) × (B + B^T) + (1/2) × (B - B^T)

Here

(1/2) × (B + B^T) is used to indicate the symmetric matrix

(1/2) × (B - B^T) is used to indicate the skew-symmetric matrix

In the following way, we can calculate the addition of a symmetric matrix and skew-symmetric matrix:

Here

is used to indicate the symmetric matrix and

is used to indicate the skew-symmetric matrix.

Difference between Symmetric matrix and Skew symmetric matrix

There is a close relation between the symmetric and skew-symmetric matrices. We can show one major difference between the symmetric matrix and the skew-symmetric matrix. This difference is described as follows:

Important Points on Symmetric matrix

There are some important points which we should know while learning the concept of a non-singular matrix. These points are described as follows:

• If the square matrix and its transpose matrix are similar to each other, then this matrix will be known as the symmetric matrix.
• We will get a symmetric matrix as a resultant matrix if we try to add the two symmetric matrices.
• The square matrix is used to contain zero elements in all off-diagonal places. That's why every square diagonal matrix will be known as symmetric.
• If any diagonal matrix and its transpose matrix are similar to each other, in this case, these types of matrices will become automatically symmetric.

Examples of Symmetric matrix

There are a lot of examples of a symmetric matrix, and some of them are described as follows:

Example 1: In this example, we have two matrices, and we have to determine which matrix is a symmetric matrix.

Solution: Here, we will first check whether matrix A is a symmetric matrix or not

Here we have A^T = -A.

Hence, matrix A is not a symmetric matrix. This property satisfies the condition of a skew-symmetric matrix. Hence this matrix is not a symmetric matrix, but it is a skew-symmetric matrix.

Now we will check the second matrix B in the following way:

Here we have B^T = B.

Hence matrix B is a symmetric matrix.

Answer: Matrix A is a skew-symmetric matrix, and matrix B is a symmetric matrix.

Example 2: In this example, we have two matrices, and we have to determine whether the given matrices are symmetric matrices or not.

Solution: Here, we will first check whether matrix M is a symmetric matrix or not. So for this, we will do the transpose of matrix M and get the following:

Here we can see that the matrix M and its transpose matrix M^T are not similar to each other. Hence matrix M is not a symmetric matrix.

Now we will check whether matrix P is a symmetric matrix. So we will again check the transpose of matrix P like this:

Here we can see that the matrix P and its transpose matrix P^T are not similar to each other. Hence matrix P is not a symmetric matrix.

Answer: Matrix M and matrix P both are not symmetric matrix.

Example 3: In this example, we have a matrix A, and the elements of matrix A are described as follows:

Here we have to determine which option is correct for matrix A.

1. Symmetric matrix
2. Skew symmetric matrix
3. Symmetric and skew-symmetric matrix
4. None of the above

Solution: The matrix A and its transpose are described as follows:

Here we can see that matrix A and its transpose matrix A^T are similar to each other. Hence matrix A is a symmetric matrix.

Answer: The correct option is (a).

Example 4: In this example, we have a symmetric matrix A, where

Now we have to determine the values of a and b.

Solution: From the question, we know that A is a symmetric matrix. So A^T = A.

Now we will compare the corresponding elements in the following way:

a + 2 = 1 ⇒ a = -1

b - 3 = 2 ⇒ b = 5

Answer: The value of a is -1, and the value of b is 5.