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In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, an operator is a function which operates on (or modifies another function 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. In Mathematics, the real numbers may be described informally in several different ways If it does, then we are not using f on its original function domain any longer. In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined 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. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally 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 Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally 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. In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. Furthermore, it is a nontrivial ideal: let n be the finite dimension of the algebra of matrices, then {I, T, T2. . . Tn} is linearly dependent. So ∑ αi Ti = 0 for some scalars αi. This implies that the polynomial ∑ αi xi lies in the ideal. Since the ring of polynomials is a principal ideal domain, this ideal is generated by some polynomial m. In Abstract algebra, a principal ideal domain, or PID is an Integral domain in which every ideal is principal i The polynomial m is precisely the minimal polynomial of T. In Linear algebra, the minimal polynomial of an n -by- n matrix A over a field F is the Monic polynomial 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. In Mathematics, and in particular Functional analysis, the shift operators are examples of Linear operators important for their simplicity and natural occurrence 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, 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

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