Prove: Every square matrix A can be written as A= A1 + A2 where A1 is Hermitian and A2 is skew- Hermitian

Question

 Matrix question

Answer 1

Let A be a square matrix and let A* denote the adjoint of A. Set A₁ = (A + A*)/2 and A₂ = (A - A*)/2. Note A₁ + A₂ = A. Since (A*)* = A, then

 

(A₁)* = (A* + A)/2 = A₁ 

 

(A₂)* = (A* - A)/2 = -(A - A*)/2 = -A₂ 

 

Therefore A₁ is Hermitian and A₂ is skew-Hermitian.