In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a functional calculus is a theory allowing one to apply
mathematical function
In mathematics, a function from a set (mathematics), set to a set assigns to each element of exactly one element of .; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously. ...
s to
mathematical operators. It is now a branch (more accurately, several related areas) of the field of
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
, connected with
spectral theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operator (mathematics), operators in a variety of mathematical ...
. (Historically, the term was also used synonymously with
calculus of variations
The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions
and functional (mathematics), functionals, to find maxima and minima of f ...
; this usage is obsolete, except for
functional derivative
In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on ...
. Sometimes it is used in relation to types of
functional equations, or in logic for systems of
predicate calculus
Predicate or predication may refer to:
* Predicate (grammar), in linguistics
* Predication (philosophy)
* several closely related uses in mathematics and formal logic:
**Predicate (mathematical logic)
** Propositional function
**Finitary relation, ...
.)
If
is a function, say a numerical function of a
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
, and
is an operator, there is no particular reason why the expression
should make sense. If it does, then we are no longer using
on its original
function domain. In the tradition of
operational calculus
Operational calculus, also known as operational analysis, is a technique by which problems in Mathematical Analysis, analysis, in particular differential equations, are transformed into algebraic problems, usually the problem of solving a polynomia ...
, algebraic expressions in operators are handled irrespective of their meaning. This passes nearly unnoticed if we talk about 'squaring a matrix', though, which is the case of
and
an
matrix
Matrix (: matrices or matrixes) or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics), a rectangular array of numbers, symbols or expressions
* Matrix (logic), part of a formula in prenex normal form
* Matrix (biology), the m ...
. 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 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
. This family is an
ideal in the ring of polynomials. Furthermore, it is a nontrivial ideal: let
be the finite dimension of the algebra of matrices, then
is linearly dependent. So
for some scalars
, not all equal to 0. This implies that the polynomial
lies in the ideal. Since the ring of polynomials is a
principal ideal domain
In mathematics, a principal ideal domain, or PID, is an integral domain (that is, a non-zero commutative ring without nonzero zero divisors) in which every ideal is principal (that is, is formed by the multiples of a single element). Some author ...
, this ideal is generated by some polynomial
. Multiplying by a unit if necessary, we can choose
to be monic. When this is done, the polynomial
is precisely the
minimal polynomial of
. This polynomial gives deep information about
. For instance, a scalar
is an eigenvalue of
if and only if
is a root of
. Also, sometimes
can be used to calculate the
exponential of
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
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operator (mathematics), operators in a variety of mathematical ...
, since for a
diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagon ...
or
multiplication operator, it is rather clear what the definitions should be.
See also
*
*
*
*
References
*
External links
*
{{DEFAULTSORT:Functional Calculus