A Hermitian matrix (or self-adjoint matrix) equals its own conjugate transpose. All eigenvalues of a Hermitian matrix are real.
⚠Hermitian matrices are named after Charles Hermite. They are fundamental in quantum mechanics where observables are represented by Hermitian operators.
What is a Hermitian Matrix?
A Hermitian matrix is a complex square matrix that equals its own conjugate transpose (A = A*). For real matrices, this reduces to a symmetric matrix. Hermitian matrices have real eigenvalues and can be diagonalized by unitary transformations.
Real Eigenvalues
All eigenvalues are real numbers, not complex.
Spectral Theorem
Can be diagonalized by unitary transformation.
Orthogonal Basis
Eigenvectors corresponding to distinct eigenvalues are orthogonal.
Real Diagonal
A = UDU* where D is real diagonal and U is unitary.
Example: A = [[2, 1+i], [1-i, 3]]
1. A* = [[2, 1-i], [1+i, 3]] (conjugate transpose)
2. A = A* ✓
3. A is Hermitian ✓
Applications
Quantum MechanicsOptimizationSignal ProcessingVibration AnalysisControl Theory
Frequently Asked Questions
What is a Hermitian matrix?▼
Matrix equal to its conjugate transpose: A = A*. For real matrices, Hermitian = symmetric. Eigenvalues are always real.
Hermitian vs symmetric?▼
Symmetric: Aᵀ = A (real case). Hermitian: A* = A (complex case). Hermitian generalizes symmetric to complex matrices.
Properties of Hermitian matrices?▼
Real eigenvalues, orthogonal eigenvectors, spectral theorem, can be diagonalized by unitary transformation.
How to check Hermitian property?▼
Verify A[i][j] = conj(A[j][i]) for all i,j. Diagonal elements must be real.
Free online calculators and tools covering mathematics, unit conversion, text processing, and daily life. Accurate, fast, mobile-friendly, and completely free to use.