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Vikram Singh
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Updated on Apr 12, 2024 10:44 IST

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.

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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## 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 = AT

### Representation:

If A=[aij]  is a symmetric matrix, then aij=aji, where:

• 1<= i <= n, 1 <= j <= n, where n belongs to natural number.
• aij is an element at the position (i, j), which is ith row and jth column
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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 = AT, B = BT, and C = CT

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 AAT=ATA.
11. ATA 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+AT will be a symmetric.

Given: A is a square matrix of real numbers

To Prove: A + AT 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 +AT………..(1)

Now, taking the transpose

BT = (A+AT)T = AT + (AT)T = AT+A = A+AT = B

Hence, B = A + AT 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 = kAT, where k is a scalar
• A is a skew-symmetric matrix iff AT = -A.

Proof:

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

A = A/2+A/2

Adding and subtracting AT on the right side, we get

A = 1/2(A + AT)+1/2(A – AT)

Let P = 1/2(A+AT) and Q = 1/2(A – AT)

Now, we will transpose of matrix P and Q

PT = 1/2(A + AT)T

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

= 1/2[AT + A]

Hence, PT = P

and,

QT = 1/2(A – AT)

= 1/2[(A)T – (AT)T

= 1/2[AT – A]

Hence QT = -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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## FAQs

What is Symmetric Matrix?

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 = transpose (A).

How do you know given matrix is symmetric or not?

It's a two-step process to check, whether a matrix is symmetric or not: 1. Given, matrix must be a square matrix. 2. Given matrix must satisfy the condition: A = transpose (A).

What are the some basic properties of a symmetric matrix?

Some basic properties of a symmetric matrices are: 1. Sum and difference of two symmetric matrix is always symmetric. 2. If A and B are symmetric matrix, then AB + BA is also symmetric. 3. Any power of symmetric matrix is symmetric.

Which type of matrices are mainly used in data science and machine learning?

In data science and machine learning, mostly real and symmetric matrices are used.