In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This concept of diagonalization is relatively straightforward for operators on finite-dimensional spaces, but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modelled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective.
Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces.
The spectral theorem also provides a canonical decomposition, called the spectral decomposition, eigenvalue decomposition, or eigendecomposition, of the underlying vector space on which the operator acts.
In this article we consider mainly the simplest kind of spectral theorem, that for a self-adjoint operator on a Hilbert space. However, as noted above, the spectral theorem also holds for normal operators on a Hilbert space.
Other articles related to "spectral theorem, spectral, theorem":
... There is also a spectral theorem for self-adjoint operators that applies in these cases ... In general, spectral theorem for self-adjoint operators may take several equivalent forms ... Spectral theorem in the form of multiplication operator ...
... In physics, particularly in quantum mechanics, the spectral theorem is expressed in a way which combines the spectral theorem as stated above and the Borel functional calculus using Dirac ... measurements are described using the spectral measure of, if the system is prepared in prior to the measurement ... If f=1, the theorem is referred to as resolution of unity In the case is the sum of an Hermitian H and a skew-Hermitian (see skew-Hermitian matrix) operator, one defines the biorthogonal basis set ...
... Theorem ... By the spectral theorem for compact self-adjoint operators, H has an orthonormal basis consisting of eigenvectors ψn of T with Tψn = μn ψn, where ...
... Theorem ... previously shown how part of Weyl's work could be simplified using von Neumann's spectral theorem.) In fact for T =D−1 with 0 ≤ T ≤ I, the spectral projection E(λ) of T is defined by It is also the ... and ψλ(1)=χλ, it follows that This identity is equivalent to the spectral theorem and Titchmarsh–Kodaira formula ...
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