Functional calculus
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2008-03-18T02:27:43Z
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In [[mathematics]], a '''functional calculus''' is a theory allowing one to apply [[mathematical function]]s to [[mathematical operator]]s. If ''f'' is a function, say a numerical function of a [[real number]], and ''M'' is an operator, there is no particular reason why the expression
:''f''(''M'')
should make sense. If it does, then we are not using ''f'' on its original [[function domain]] any longer. This passes nearly unnoticed if we talk about 'squaring a matrix', though, which is the case of ''f''(''x'') = ''x''<sup>2</sup> and ''M'' an ''n''×''n'' [[matrix (mathematics)|matrix]]. The idea of a functional calculus is to create a ''principled'' approach to this kind of [[overloading]] of the notation.
The most immediate case is to apply [[polynomial function]]s to a [[square matrix]], extending what has just been discussed. In the finite dimensional case, the polynomial functional calculus yields quite a bit of information about the operator. For example, consider the family of polynomials which annihilates an operator ''T''. This family is an [[ideal (ring theory)|ideal]] in the ring of polynomials. Furthermore, it is a nontrivial ideal: let ''n'' be the finite dimension of the algebra of matrices, then {''I'', ''T'', ''T''<sup>2</sup>...''T<sup>n</sup>''} is linearly dependent. So ∑ ''α<sub>i</sub> T<sup>i</sup>'' = 0 for some scalars ''α<sub>i</sub>''. This implies that the polynomial ∑ ''α<sub>i</sub> x<sup>i</sup>'' lies in the ideal. Since the ring of polynomials is a [[principal ideal domain]], this ideal is generated by some polynomial ''m''. The polynomial ''m'' is precisely the [[minimal polynomial]] of ''T''. One has, for instance, a scalar ''α'' is an eigenvalue of ''T'' if and only if ''α'' is a root of ''m''. Also, sometimes ''m'' can be used to calculate the [[exponential]] of ''T'' efficiently.
The polynomial calculus is not as informative in the infinite dimensional case. Consider the [[unilateral shift]] with the polynomials calculus; the ideal defined above is now trivial. Thus one is interested in functional calculi more general than polynomials. The subject is closely linked to [[spectral theory]], since for a [[diagonal matrix]] or [[multiplication operator]], it is rather clear what the definitions should be.
For technical accounts see:
*[[holomorphic functional calculus]]
*[[continuous functional calculus]]
*[[Borel functional calculus]].
[[Category:Functional calculus| ]]
[[nl:Functionele calculus]]