Symmetric operator

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

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