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In mathematics, a self-adjoint operator on a complex vector space with inner product is a linear map that is its own adjoint. That is, for all . If is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of is a Hermitian matrix, i.e., equal to its conjugate transpose . By the finite-dimensional spectral theorem, has an orthonormal basis such that the matrix of 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.
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