Chapter 1.7, Problem 44E

In the text we defined a matrix A to be symmetric if A^T = A. Analogously, a matrix A is said to be skew-symmetric if A^T = −A. Exercises 41–45 are concerned with matrices of this type.

Prove that every square matrix A can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix. [Hint: Note the identity $A=\frac{1}{2}(A+A^T)+\frac{1}{2}(A-A^T)$.]