ickehermitiska

Ickehermitiska, or non-Hermitian, refers to matrices or operators that are not equal to their Hermitian conjugate. An operator A is ickehermitisk if A does not equal A†. Non-Hermitian operators can have complex eigenvalues, and their eigenvectors need not be orthogonal. In general, the eigenvectors of A and of A† form a biorthogonal set, and A may be diagonalizable with a full set of eigenvectors or defective, featuring Jordan blocks.

In many applications, ickehermitiska operators describe open or dissipative systems where probability or norm is not strictly conserved. Examples include gain and loss in optical systems, decaying states in quantum mechanics, or effective Hamiltonians used to model open quantum systems. Two notable features are exceptional points, parameter values where both eigenvalues and eigenvectors coalesce, and the resulting high sensitivity of spectra to perturbations. Non-Hermitian systems can also exhibit parity–time (PT) symmetry, which in some regimes yields real eigenvalues despite non-Hermiticity.

Mathematically, analysis of ickehermitiska operators often involves concepts such as the pseudospectrum and biorthogonal bases. The spectral theorem for Hermitian operators does not apply, so researchers use tools like singular value decomposition, non-Hermitian perturbation theory, and specialized stability analyses to understand their behavior.

Common examples include simple 2x2 matrices with asymmetric entries, effective Hamiltonians in scattering theory, and photonic lattices with balanced gain and loss. The study of ickehermitiska systems spans linear algebra, quantum mechanics, and wave physics, reflecting the broad presence of non-conservative dynamics in science and engineering.