All about Symmetric Matrix

A square matrix in Linear algebra is said to be a symmetric matrix if it is equal to its transpose. In this article, we will briefly discuss symmetric matrix, its properties and theorems related to symmetric matrices.

A matrix is a rectangular arrangement of numbers (real or complex) or symbols arranged in rows and columns. The number in the matrix are called the elements, and if the matrix has m rows and n columns, then the matrix is said to be an “m by n” matrix. If the number of rows and columns is equal (i.e., m = n), then the matrix is said to be a square matrix. There are different types of matrices, and in this article, we will discuss a square matrix known as Symmetric Matrix with some additional properties. 

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Now, let’s discuss one more topic before the start of the article i.e., the transpose of a matrix.

Transpose of Matrix

Transpose of a matrix is obtained by interchanging rows to column ( or column to rows), i.e., if there are m rows and n columns, then the transpose matrix will have n rows and m columns.

It is represented by T.

Let’s take an example:

Example 1:

B matrix is a square matrix, and elements of matrix B are complex numbers.

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Table of Content

• What is a Symmetric Matrix?
• Properties
• Theorems

What is a Symmetric Matrix?

Definition

A square matrix that is equal to its transpose is called a symmetric matrix, i.e., if A is any square matrix, then A is said to be a symmetric matrix if and only if:

A = A^T

Representation:

If A=[a_ij]  is a symmetric matrix, then a_ij=a_ji, where:

• 1<= i <= n, 1 <= j <= n, where n belongs to natural number.
• a_ij is an element at the position (i, j), which is i^th row and j^th column

Now, let’s take some example to get a better understanding:

Examples:

Here, we have taken three examples of square matrices of different orders, and when we look closely, we will get:

A = A^T, B = B^T, and C = C^T

Thus, A, B, and C are all symmetric matrices.

Note: The matrix C is also known as the Identity Matrix of order 3.

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Properties of Symmetric Matrix

1. The sum of a symmetric matrix is a symmetric.
2. When a symmetric matrix is multiplied by a scalar, the result will be a symmetric.
3. If A and B are two symmetric matrices, then AB + BA is also symmetric.
4. If A is a symmetric matrix, then any power of A (i.e., Aⁿ, where n is a Natural Number) will be Symmetric.
5. If A is a symmetric matrix as well as invertible, then the inverse of A will also be Symmetric.
6. Every diagonal matrix is a symmetry matrix.
7. Symmetric matrices have real eigenvalues and are always diagonalizable.
8. Eigenvectors corresponding to distinct eigenvalues are Orthogonal.
9. If a symmetric matrix is positive definite, then all eigen values will be positive.
10. If A is a symmetric matrix then AA^T=A^TA.
11. A^TA is invertible if and only if the columns of A are linearly independent.

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Symmetric Matrix Theorems: 

Theorem 1: If  A be any square matrix having real elements, then A+A^T will be a symmetric.

Given: A is a square matrix of real numbers

To Prove: A + A^T is symmetric

Concepts to be Used:

Transpose Matrix Properties

Transpose Matrix Properties

• Transpose of a matrix is the matrix itself
• Sum of transpose of matrices is equal to the transpose of the sum of two matrices

Proof:

Let B be any matrix, such that

B = A +A^T………..(1)

Now, taking the transpose

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

Hence, B = A + A^T is a symmetric.

Theorem 2: Every square matrix can be decomposed uniquely as the sum of symmetric and skew symmetric matrix.

Given: A is a square matrix

To Show: Any square matrix cabe expressed as a sum of symmetric and skew symmetric matrices.

Concept to be used:

• Transpose of a matrix is the matrix itself
• Sum of transpose of matrices is equal to the transpose of the sum of two matrices
• (kA)^T = kA^T, where k is a scalar
• A is a skew-symmetric matrix iff A^T = -A.

Proof: 

Let A be any square matrix, that can be written as:

A = A/2+A/2

Adding and subtracting A^T on the right side, we get

A = 1/2(A + A^T)+1/2(A – A^T)

Let P = 1/2(A+A^T) and Q = 1/2(A – A^T)

Now, we will transpose of matrix P and Q

P^T = 1/2(A + A^T)^T 

= 1/2[(A)^T + (A^T)^T]

= 1/2[A^T + A]

Hence, P^T = P

and, 

Q^T = 1/2(A – A^T)^T 

= 1/2[(A)^T – (A^T)^T

= 1/2[A^T – A]

Hence Q^T = -Q

So, P is a symmetric and Q is a skew-symmetric matrix and A is the sum of P and Q.

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Conclusion

In this article, we have discussed symmetric matrix, its properties and theorems and some of the examples. Hope you will like the article.
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