A square matrix is said to be a symmetric matrix if for all values of i and j or we can say that

Example of symmetric matrix is-

And if a square matrix is called skew symmetric matrix if it follows the following conditions-

- for all values of i and j.
- The diagonal elements of the matrix should be zero

For example-

**Note-**

We can express any square matrix as the sum of two matrices, where one is symmetric and the other one is anti-symmetric.

So that-

**Square matrix = Symmetric matrix + Anti-symmetric matrix**

**Important points about symmetric matrix- **

- Eigen vectors of a symmetric matrix corresponding to different Eigen values are orthogonal.

- The product of two symmetric matrices A and B is symmetric if AB = BA.

- Inverse of a non-singular symmetric matrix A is symmetric.
- We can reduce a real symmetric matrix A to a diagonal form N’A N = D, here N is the normalized orthogonal modal matrix of A and D is called spectoral matrix.

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