Hermitian plus skew-Hermitian decomposition

A complex square matrix A can be written as A = P + Q, where P = ½(A + A*) is Hermitian and Q = ½(A − A*) is skew-Hermitian, and the decomposition is unique. Here A* = (Ā)ᵀ is the conjugate transpose.

A matrix H is Hermitian if H* = H, and skew-Hermitian if H* = −H.

Existence

Define P = ½(A + A*) and Q = ½(A − A*). Using (A*)* = A and (kA)* = k̄A* with k = ½ real:

• P* = ½(A* + A) = P, so P is Hermitian.
• Q* = ½(A* − A) = −Q, so Q is skew-Hermitian.
• P + Q = A.

Uniqueness

Suppose A = R + S with R Hermitian and S skew-Hermitian. Taking the conjugate transpose:

A* = R* + S* = R − S

Adding and subtracting with A = R + S:

A + A* = 2R  ⇒  R = ½(A + A*) = P
A − A* = 2S  ⇒  S = ½(A − A*) = Q

So the decomposition is unique.

Example. A = [1 2+i; 3 i]. Then A* = [1 3; 2−i −i]. P = ½(A + A*) = [1 (5+i)/2; (5−i)/2 0] (Hermitian), and Q = ½(A − A*) = [0 (−1+i)/2; (1+i)/2 i] (skew-Hermitian), with P + Q = A. ∎