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Hermitian and skew-Hermitian matrices

Short answer

Hermitian: A* = A; skew-Hermitian: A* = −A. Then (A+A*)* = A*+A = A+A*, so Hermitian; (A−A*)* = A*−A = −(A−A*), so skew-Hermitian.

Define Hermitian and skew-Hermitian matrices of a square complex matrix. If A is any square matrix, prove that A + A* is Hermitian and A − A* is skew-Hermitian.

Definitions

Let A* (also written Aᶿ) denote the conjugate transpose of A: A* = (Ā)ᵀ, i.e. take the complex conjugate of every entry and transpose.

  • A square matrix A is Hermitian if A* = A, i.e. aᵢⱼ = conj(aⱼᵢ). Its diagonal entries are real. Example: [2 1−i; 1+i 3].
  • A square matrix A is skew-Hermitian if A* = −A, i.e. aᵢⱼ = −conj(aⱼᵢ). Its diagonal entries are purely imaginary or zero. Example: [i 2+i; −2+i 0].

Proof that A + A* is Hermitian and A − A* is skew-Hermitian

We use the properties (A + B)* = A* + B*, (A − B)* = A* − B*, and (A*)* = A.

A + A is Hermitian:*

(A + A*)* = A* + (A*)* = A* + A = A + A*

So (A + A*)* = A + A*, i.e. A + A* is Hermitian.

A − A is skew-Hermitian:*

(A − A*)* = A* − (A*)* = A* − A = −(A − A*)

So (A − A*)* = −(A − A*), i.e. A − A* is skew-Hermitian. ∎

Consequence: A = ½(A + A*) + ½(A − A*), which expresses A as the sum of a Hermitian and a skew-Hermitian matrix.

✓ Answer checked by Claude Opus 5.5 (high effort)

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