Diagonal matrix in the context of Spectral theorem

In mathematics, a self-adjoint operator on a complex vector space ${\displaystyle V}$ with inner product ${\displaystyle \langle \cdot ,\cdot \rangle }$ is a linear map ${\displaystyle A}$ (from ${\displaystyle V}$ to itself) that is its own adjoint. That is, ${\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle }$ for all ${\displaystyle x,y\in V}$. If ${\displaystyle V}$ is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of ${\displaystyle A}$ is a Hermitian matrix, i.e., equal to its conjugate transpose ${\displaystyle A^{*}}$. By the finite-dimensional spectral theorem, ${\displaystyle V}$ has an orthonormal basis such that the matrix of ${\displaystyle A}$ relative to this basis is a diagonal matrix with entries in the real numbers. This article deals with applying generalizations of this concept to operators on Hilbert spaces of arbitrary dimension.

Self-adjoint operators are used in functional analysis and quantum mechanics. In quantum mechanics their importance lies in the Dirac–von Neumann formulation of quantum mechanics, in which physical observables such as position, momentum, angular momentum and spin are represented by self-adjoint operators on a Hilbert space. Of particular significance is the Hamiltonian operator ${\displaystyle {\hat {H}}}$ defined by