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Jitse Niesen (talk | contribs) →Spectral theorem: revert - P is applied to T, not to Tv. Also, be pedantic: the formula is not a P(T), but P(T)v |
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where <math>\mathbf{v}_1, \mathbf{v}_2, \dots</math> and <math>\lambda_1, \lambda_2, \dots</math> stand for the eigenvectors and eigenvalues of <math>\mathcal{T}</math>. The simplest case in which the theorem is valid is the case where the linear transformation is given by a [[real number|real]] [[symmetric matrix]] or [[complex number|complex]] [[Hermitian matrix]]; more generally the theorem holds for all [[normal matrix|normal matrices]].
If one defines the ''n''th power of a transformation as the result of applying it ''n'' times in succession, one can also define [[polynomial]]s of transformations. A more general version of the theorem is that any polynomial ''P'' of <math>\mathcal{T}</math> is
:<math>P(\mathcal{T})(\mathbf{v})
The theorem can be extended to other functions of transformations like [[analytic function]]s, the most general case being [[Measurable function|Borel functions]].
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