Question

prove that any square Matrix can be expressed as the sum of a symmetric and skew symmetric matrix​

Answer 1

Answer:

Yes. Every square matrix can be expressed as a sum of a symmetric matrix and a skew symmetric matrix.

Let A be a square matrix of order n. Then A can be written as follows:

A= [(A+A’)/2 + (A-A’)/2]

Where A’ is the transpose of the matrix A.

(A+A’)/2 is a symmetric matrix and (A-A’)/2 is a skew symmetric matrix(please check it). Also, this representation is unique