Editing Spectral theorem (section)

{{Short description|Result about when a matrix can be diagonalized}}
In [[linear algebra]] and [[functional analysis]], a '''spectral theorem''' is a result about when a [[linear operator]] or [[matrix (mathematics)|matrix]] can be [[Diagonalizable matrix|diagonalized]] (that is, represented as a [[diagonal matrix]] in some basis). This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on [[finite-dimensional vector space]]s but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of [[linear operator]]s that can be modeled by [[multiplication operator]]s, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative [[C*-algebra]]s. See also [[spectral theory]] for a historical perspective.

Examples of operators to which the spectral theorem applies are [[self-adjoint operator]]s or more generally [[normal operator]]s on [[Hilbert space]]s.

The spectral theorem also provides a [[canonical form|canonical]] decomposition, called the '''[[eigendecomposition of a matrix|spectral decomposition]]''', of the underlying vector space on which the operator acts.

[[Augustin-Louis Cauchy]] proved the spectral theorem for [[Symmetric matrix|symmetric matrices]], i.e., that every real, symmetric matrix is diagonalizable. In addition, Cauchy was the first to be systematic about [[determinant]]s.<ref>{{cite journal| doi=10.1016/0315-0860(75)90032-4 | volume=2 | title=Cauchy and the spectral theory of matrices | year=1975 | journal=Historia Mathematica | pages=1–29 | last1 = Hawkins | first1 = Thomas| doi-access=free }}</ref><ref>[http://www.mathphysics.com/opthy/OpHistory.html A Short History of Operator Theory by Evans M. Harrell II]</ref> The spectral theorem as generalized by [[John von Neumann]] is today perhaps the most important result of [[operator theory]].

This article mainly focuses on 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.