: Mathématiques: spectral theorem

mathematical analysis > functional analysis > spectral theory > spectral theorem

algebra > linear algebra > spectral theory > spectral theorem

mathematical analysis > functional analysis > operator algebra > spectral theorem

... > abstract algebra > algebraic structure > ring theory > operator algebra > spectral theorem

category theory > morphism > endomorphism > spectral theorem

algebra > abstract algebra > morphism > endomorphism > spectral theorem

spectral theorem

In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be 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 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 modeled 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.

(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Spectral_theorem)

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