A Hermitian matrix or operator is a mathematical object that equals its own complex conjugate transpose .
Hermitian Matrix
A square matrix where the entry in the i-th row and j-th column is the complex conjugate of the entry in the j-th row and i-th column.
Hermitian Operator
A linear operator on an inner product space that equals its own adjoint (Conjugate Transpose). These operators are foundational in physics. They represent measurable physical quantities, known as observables, like position, momentum, and energy.
Properties
- The eigenvalues of any Hermitian matrix or operator are always real numbers. This matches real-world measurements in physics.
- Eigenvectors that belong to different eigenvalues are always orthogonal.