**Define Symmetric Matrix with Examples**

In linear algebra, A square matrix A is called a** symmetric matrix** if we take a transpose of a matrix and the answer is itself matrix.

** such that Symmetric Matrix**

**Symmetric Matrix Example 3×3**

A^t = A

A symmetric **matrix** is always a **square matrix** because equal matrices have equal dimension. In the symmetric matrix, the number of rows and the number of columns are the same as it is a square matrix. Every real symmetric matrix can be diagonalizable.

** General form of Symmetric matrix **

A symmetric matrix is important and useful in many applications because of its application. Some examples of well-known symmetric matrices are the correlation matrix covariance matrix and distance matrix.

*Before we move further first we discuss that what is a square matrix is and what is the transpose of a matrix*

A matrix is said to be a square matrix if and only if the number of columns of rows is equal to the number of columns. A number of rows shown by m and number of columns are shown by n. in square matrix m = n.

**Transpose of a ****symmetric matrix**

**symmetric matrix**

If we want to take the transpose of any matrix then we interchanging the rows and columns of the original matrix. If a matrix A has m × n order and we take transpose then the order is changing and the order is n × m.

**Example of symmetric matrix **

A=\;\begin{bmatrix}1&2\\2&1\end{bmatrix}
order of this matrix **A** is 2 by 2

Firstly we change the first row to the first column

Then we change the second row to the second column

**Example of a symmetric matrices**

A=\;\begin{bmatrix}3&-2&4\\-2&6&2\\4&2&3\end{bmatrix}
Here the order of the matrix A is 3 -by 3.

**Properties of symmetric matrices**

- The product of two symmetric matrices may not be symmetric. That is if

A^t = A , B^t = B then (AB)^t = B^t A^t = BA not equal to AB is general.it is only possible when A and B are commute like AB = BA

- The sum of two symmetric matrices is again a symmetric matrices.
- The difference of two symmetric matrices is also again a symmetric matrice.
- The most important property of symmetric matrices is that their eigenvalues behave very nicely.
- Hermitian matrice is also symmetric matrix if the entries are complex number.
- If all eigenvalues of symmetric matrix A are different then the matrix A can transformed into its diagonal matrice.
- There is no complex number in Eigen value of a symmetric matrix that is means there are all real numbers.
- Symmetric matrices has linearly independent eigen vector.

*Note* symmetric matrix

*Note*symmetric matrix

The zero matrix is also a symmetric matrix.