prove any square matrix can be expressed as the sum of symmetric and skew symmetric matrix - Maths - Matrices

Question

prove any square matrix can be expressed as the sum of symmetric
and skew symmetric matrix?

Manbar Singh · Accepted Answer

Let A be any square matrix
Then we can say,A = 12A+A' + 12A-A' = P + Q, where P = 12A+A'; Q = 12A-A'Now, P' = 12A+A''⇒P' = 12A+A''⇒P' = 12A' + A''⇒P' = 12A'+A⇒P'=12A+A' = PThus, P' = P and so P is symmetric
Now, Q' = 12A-A''⇒Q' = 12A-A''⇒Q' = 12A' - A''⇒Q' = 12A'-A⇒Q'=-12A-A' =-QThus, Q' = -Q and so Q is skew symmetric
So, A = P+Q, where P is symmetric and Q is skew symmetric

prove any square matrix can be expressed as the sum of symmetric and skew symmetric matrix?