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 can be defined by substituting by a square matrix: powers become matrix powerss, additions become matrix sums and multiplications by coefficients become scalar multiplications, then the corresponding matrix series converges for matrices for some matrix norm.

Diagonalizable matrices

A square matrix is diagonalizable, if there is an invertible matrix is a diagonal matrix, that is,

It can be verified that the matrix does not depend on a particular choice of .

Jordan decomposition

All complex matrices, whether they are diagonalizable or not, have a Jordan normal form, where the matrix J consists of Jordan blocks. Proper method is presented in details in section 1.2.2, pages 4-7, of the book Higham (2008) by the Hermite interpolation with the use of confluent Vandermonde matrix. Consider these blocks separately and apply the power series

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:

is a positive-semidefinite matrix.) 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 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,

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

, which has a finite length Taylor series. We compute this in two ways,

. For some f(x. The expansion of the first term then follows the power series given above,

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 are the eigenvalues of its characteristic equation,

Examples

• Matrix polynomial

• Matrix root

• Matrix logarithm

• Matrix exponential

• Matrix sign function

Classes of matrix functions

Using the semidefinite ordering is positive-semidefinite is positive definite), some of the classes of scalar functions can be extended to matrix functions of Hermitian matrices.

Operator monotone

A function 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 for all self-adjoint matrices with spectra in the domain of . This definition is analogous to a concave scalar function. An operator convex function 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.

See also

• Algebraic Riccati equation

• Sylvester's formula

• Loewner order

• Matrix calculus

• Trace inequalities

• Trigonometric functions of matrices

References

Further reading