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?**

**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:**

**Representation:**

If **A**=[*a _{ij}*] is a symmetric matrix, then

*a*, where:

_{ij}=a_{ji}- 1<=
*i*<= n, 1 <=*j*<= n, where*n*belongs to natural number. *a*is an element at the position (i, j), which is_{ij}*i*row and^{th}*j*column^{th}

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

**Examples:**

**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**

- The sum of a symmetric matrix is a symmetric.
- When a symmetric matrix is multiplied by a scalar, the result will be a symmetric.
- If A and B are two symmetric matrices, then AB + BA is also symmetric.
- If A is a symmetric matrix, then any power of A (i.e., A
^{n}, where n is a Natural Number) will be Symmetric. - If A is a symmetric matrix as well as invertible, then the inverse of A will also be Symmetric.
- Every diagonal matrix is a symmetry matrix.
- Symmetric matrices have real eigenvalues and are always diagonalizable.
- Eigenvectors corresponding to distinct eigenvalues are Orthogonal.
- If a symmetric matrix is positive definite, then all eigen values will be positive.
- If A is a symmetric matrix then AA
^{T}=A^{T}A. - A
^{T}A 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.

^{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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## 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.

**About the Author**

Vikram has a Postgraduate degree in Applied Mathematics, with a keen interest in Data Science and Machine Learning. He has experience of 2+ years in content creation in Mathematics, Statistics, Data Science, and Mac... Read Full Bio

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