Functional calculus
In mathematics, a
functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch of the field of functional analysis, connected with spectral theory. 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 should make sense. If it does, then we are no longer using
f on its original function domain. In the tradition of operational calculus, 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
f =
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. 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.