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In mathematics, the functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. 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 Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ...
In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...
This article is about operators in mathematics, for other kinds of operators see operator (disambiguation). ...
In mathematics, the real numbers may be described informally in several different ways. ...
- 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) = x2 and M an n×n matrix. The idea of a functional calculus is to create a principled approach to this kind of overloading of the notation. In mathematics, the domain of a function is the set of all input values to the function. ...
In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ...
In computer science, overloading is a type of polymorphism where different functions with the same name are invoked based on the data types of the parameters passed. ...
The most immediate case is to apply polynomial functions 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 in the ring of polynomials. Furthermore, by the Cayley-Hamilton theorem, it is nontrivial. Since the ring of polynomials is a principal ideal domain, the ideal is generated by some polynomial m. The polynomial m is precisely the minimal polynomial of T, and it can be used to calculate, for example, 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. In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...
For the square matrix section, see square matrix. ...
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ...
In linear algebra, the Cayley-Hamilton theorem (named after the mathematicians Arthur Cayley and William Hamilton) states that every square matrix over the real or complex field, satisfies its own characteristic equation. ...
In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ...
In mathematics, the minimal polynomial of an object α is the monic polynomial p of least degree such that p(α)=0. ...
The term exponential may refer to any of several topics in mathematics: Exponential distribution Exponential function Exponential growth, exponential decay Exponential time Matrix exponential Exponential map (in differential geometry) All relate in some fashion to exponents. ...
In mathematics, and in particular functional analysis, the shift operators are examples of linear operators, important for their simplicity and natural occurrence. ...
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix. ...
In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. ...
In operator theory, a multiplication operator is a linear operator T defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f. ...
For technical accounts see:
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