Question

Prove that every square matrix can be uniquely expressed as the sum of a symmetric matrix and a skew-symmetric matrix.

Answer 1

Let a be a square matric.
Then, `A=1/2 (A+A^(T))+1/2 (A-A^(T))`
Here, `P=1/2 (A+A^(T))` is a symmetric matrix and `Q=1/2 (A-A^(T))` is a skew-symmetric matrix.
Thus, matrix A can be expressed as sum of symmetric and skew-symmetric matrix.