# What is the difference between Symmetric vs Skew Symmetric?

I want to know the difference between Symmetric Symmetric vs Skew Symmetric?

A symmetric matrix satisfies $A^T = A$

A skew-symmetric matrix satisfies $A^T = -A$.

Additionally, it is a fact that every matrix can be written as the sum of a symmetric matrix and a skew-symmetric matrix:

$$A = \frac{1}{2}(A + A^T) + \frac{1}{2}(A - A^T)$$

then $B = \frac{1}{2}(A + A^T)$ is symmetric since

$$B^T = \frac{1}{2}(A + A^T)^T = \frac{1}{2}(A^T + A) = B$$

and $C = \frac{1}{2}(A - A^T)$ is skew-symmetric:

$$C^T = \frac{1}{2}(A - A^T)^T = \frac{1}{2}(A^T - A) = -C$$

$$\mathbf{Symmetric}: \quad M^T=M$$ $$\mathbf{Skew\text{-}symmetric}: \quad M^T=-M$$

Skew-symmetric:

$A=-A^T$

Check out this definition