Analytic function of a matrix

In mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of the same size.
This is used for defining the exponential of a matrix, which is involved in the closed-form solution of systems of linear differential equations.
Extending scalar function to matrix functions.
There are several techniques for lifting a real function to a square matrix function such that interesting properties are maintained. All of the following techniques yield the same matrix function, but the domains on which the function is defined may differ.
Power series.
If the analytic function has the Taylor expansion
then a matrix function formula_2 can be defined by substituting by a square matrix: powers become matrix powers, additions become matrix sums and multiplications by coefficients become scalar multiplications. If the series converges for formula_3, then the corresponding matrix series converges for matrices such that formula_4 for some matrix norm that satisfies formula_5.
Diagonalizable matrices.
A square matrix is diagonalizable, if there is an invertible matrix such that formula_6 is a diagonal matrix, that is, has the shape
As formula_8 it is natural to set
It can be verified that the matrix does not depend on a particular choice of .
For example, suppose one is seeking formula_10 for 
One has 
for
Application of the formula then simply yields 
Likewise, 
Jordan decomposition.
All complex matrices, whether they are diagonalizable or not,
have a Jordan normal form formula_16, where the matrix "J" consists of Jordan blocks. Consider these blocks separately and apply the power series to a Jordan block:
This definition can be used to extend the domain of the matrix function
beyond the set of matrices with spectral radius smaller than the radius of convergence of the power series.
Note that there is also a connection to divided differences.
A related notion is the Jordan–Chevalley decomposition which expresses a matrix as a sum of a diagonalizable and a nilpotent part.
Hermitian matrices.
A Hermitian matrix has all real eigenvalues and can always be diagonalized by a unitary matrix P, according to the spectral theorem.
In this case, the Jordan definition is natural. Moreover, this definition allows one to extend standard inequalities for
real functions:
If formula_18 for all eigenvalues of formula_19, then formula_20.
The proof follows directly from the definition.
Cauchy integral.
Cauchy's integral formula from complex analysis can also be used to generalize scalar functions to matrix functions. Cauchy's integral formula states that for any analytic function defined on a set , one has
where is a closed simple curve inside the domain enclosing . 
Now, replace by a matrix and consider a path inside that encloses all eigenvalues of . One possibility to achieve this is to let be a circle around the origin with radius larger than ‖‖ for an arbitrary matrix norm ‖•‖. Then, is definable by
This integral can readily be evaluated numerically using the trapezium rule, which converges exponentially in this case. That means that the precision of the result doubles when the number of nodes is doubled. In routine cases, this is bypassed by Sylvester's formula.
This idea applied to bounded linear operators on a Banach space, which can be seen as infinite matrices, leads to the holomorphic functional calculus.
Matrix perturbations.
The above Taylor power series allows the scalar formula_24 to be replaced by the matrix. This is not true in general when expanding in terms of formula_25 about formula_26 unless formula_27. A counterexample is formula_28, which has a finite length Taylor series. We compute this in two ways,
The scalar expression assumes commutativity while the matrix expression does not, and thus they cannot be equated directly unless formula_27. For some "f"("x") this can be dealt with using the same method as scalar Taylor series. For example, formula_33. If formula_34 exists then formula_35. The expansion of the first term then follows the power series given above,
The convergence criteria of the power series then apply, requiring formula_37 to be sufficiently small under the appropriate matrix norm. For more general problems, which cannot be rewritten in such a way that the two matrices commute, the ordering of matrix products produced by repeated application of the Leibniz rule must be tracked.
Arbitrary function of a 2×2 matrix.
An arbitrary function "f(A)" of a 2×2 matrix A has its Sylvester's formula simplify to 
where formula_39 are the eigenvalues of its characteristic equation, |A-λI|=0, and are given by
Classes of matrix functions.
Using the semidefinite ordering (formula_41 is positive-semidefinite and 
formula_42 is positive definite), some
of the classes of scalar functions can be extended to matrix functions of Hermitian matrices.
Operator monotone.
A function is called operator monotone if and only if formula_43 for all self-adjoint matrices with spectra in the domain of . This is analogous to monotone function in the scalar case.
Operator concave/convex.
A function is called operator concave if and only if
for all self-adjoint matrices with spectra in the domain of and formula_45. This definition is analogous to a concave scalar function. An operator convex function can be defined be switching formula_46 to formula_47 in the definition above.
Examples.
The matrix log is both operator monotone and operator concave. The matrix square is operator convex. The matrix exponential is none of these. Loewner's theorem states that a function on an "open" interval is operator monotone if and only if it has an analytic extension to the upper and lower complex half planes so that the upper half plane is mapped to itself.