Editing Diagonalizable matrix (section)

== Quantum mechanical application ==
In [[quantum mechanics|quantum mechanical]] and [[quantum chemistry|quantum chemical]] computations matrix diagonalization is one of the most frequently applied numerical processes. The basic reason is that the time-independent [[Schrödinger equation]] is an eigenvalue equation, albeit in most of the physical situations on an infinite dimensional [[Hilbert space]].

A very common approximation is to truncate (or project) the Hilbert space to finite dimension, after which the  Schrödinger equation can be formulated as an eigenvalue problem of a real symmetric, or complex Hermitian matrix. Formally this approximation is founded on the [[variational principle]], valid for Hamiltonians that are bounded from below.

[[Perturbation theory (quantum mechanics)#First order corrections|First-order perturbation theory]] also leads to matrix eigenvalue problem for degenerate states.