What is a Symmetric Matrix?

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-

1. for all values of i and j. 
2. 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- 

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

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

3. Inverse of a non-singular symmetric matrix A is symmetric.
4. 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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