Question

For any n x n matrix. Prove that A can be uniquely expressed as a sum of symmetric matrix and skew symmetric matrix.

Answer 1

Let A be any square matrix.

Now,

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

$=P+\theta$

where, $P=\frac{1}{2}(A+A).,\theta =\frac{1}{2}(A-A)$

To prove, P is symmetric i.e. $P=P$ & $\theta$ is skew - symmetric , i.e. $\theta =-\theta$

$P=\frac{1}{2}(A+A)=\frac{1}{2}(A+(A\left)\right)$

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

$=P$

So, P is symmetric

$\theta =\frac{1}{2}(A-A)=\frac{1}{2}(A-(A\left)\right)$

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

$=-\theta$

So, $\theta$ is skew - symmetric

Hence, the representation $A=P+\theta$ is unique, hence it is proved that every square matrix can be uniquely expressed as a sum of symmetric and skew symmetric matrix

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