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Erschienen in:
2009 | OriginalPaper | Buchkapitel
Square Roots and the Polar Decomposition
verfasst von : Rajendra Bhatia
Erschienen in: Notes on Functional Analysis
Verlag: Hindustan Book Agency
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One of the most important and useful theorems of linear algebra is the spectral theorem. This says that every normal operator on an n-dimensional Hilbert space ℋ can be diagonalised by a unitary conjugation: there exists a unitary operator U such that U*AU = Λ, where Λ is the diagonal matrix with the eigenvalues of A on its diagonal. Among other things, this allows us to define functions of a normal matrix A in a natural way. Let f be any functions on ℂ. If Λ = diag (λ1,…, λ n ) is a diagonal matrix with λ j as its diagonal entries, define f(Λ) to be the diagonal matrix diag (f(λ1),…,f(λ n )), and if A = UΛU*, put f(A) = Uf(Λ)U*.