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Skew-Hermitian Matrix Checker

Check if matrix is skew-Hermitian (A = -A*)

Enter 2x2 Complex Matrix (real + imaginary i)

Skew-Hermitian Matrix Definition

A is skew-Hermitian ⟺ A* = -A

A* = conjugate transpose of A

Diagonal elements must be PURE IMAGINARY

Real skew-Hermitian = skew-symmetric

A skew-Hermitian (or anti-Hermitian) matrix equals the negative of its conjugate transpose. All eigenvalues are pure imaginary or zero.

⚠ Skew-Hermitian matrices are related to Hermitian matrices: if H is Hermitian, then iH is skew-Hermitian, and vice versa.

What is a Skew-Hermitian Matrix?

A skew-Hermitian matrix is a complex square matrix A satisfying A* = -A, where A* is the conjugate transpose. The diagonal elements must be pure imaginary (zero real part), and eigenvalues are pure imaginary or zero.

Pure Imaginary Eigenvalues

All eigenvalues are purely imaginary or zero.

Pure Imaginary Diagonal

Diagonal elements have zero real part.

Relation to Hermitian

i·H is skew-Hermitian if H is Hermitian.

Unitary Similarity

Example: A = [[0, 1+i], [-1+i, 0]]
1. A* = [[0, -1-i], [1-i, 0]] (conjugate transpose)
2. -A* = [[0, 1+i], [-1+i, 0]] = A ✓
3. A is skew-Hermitian ✓

Applications

Quantum Mechanics Lie Algebras Physics Rotations Control Theory

Frequently Asked Questions

What is a skew-Hermitian matrix?▼

Matrix where A* = -A. Equivalently, A = -A*. Diagonal elements must be pure imaginary or zero.

Skew-Hermitian vs skew-symmetric?▼

Skew-symmetric: Aᵀ = -A (real case). Skew-Hermitian: A* = -A (complex case). Generalizes to complex matrices.

Properties of skew-Hermitian matrices?▼

Eigenvalues are pure imaginary or zero. Can be written as iH where H is Hermitian. Sum of skew-Hermitian matrices is skew-Hermitian.

How to check skew-Hermitian property?▼

Verify A[i][j] = -conj(A[j][i]) for all i,j. Diagonal elements must be pure imaginary (zero real part).

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