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. 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. If the analytic function f has the Taylor expansion then a matrix function can be defined by substituting x by a square matrix: powers become matrix powers, additions become matrix sums and multiplications by coefficients become scalar multiplications. If the series converges for , then the corresponding matrix series converges for matrices A such that for some matrix norm that satisfies . A square matrix A is diagonalizable, if there is an invertible matrix P such that is a diagonal matrix, that is, D has the shape As it is natural to set It can be verified that the matrix f(A) does not depend on a particular choice of P. For example, suppose one is seeking for One has for Application of the formula then simply yields Likewise, Jordan normal form All complex matrices, whether they are diagonalizable or not, have a Jordan normal form , 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. A Hermitian matrix has all real eigenvalues and can always be diagonalized by a unitary matrix P, according to the spectral theorem.

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Jordan matrix

In the mathematical discipline of matrix theory, a Jordan matrix, named after Camille Jordan, is a block diagonal matrix over a ring R (whose identities are the zero 0 and one 1), where each block along the diagonal, called a Jordan block, has the following form: Every Jordan block is specified by its dimension n and its eigenvalue , and is denoted as Jλ,n. It is an matrix of zeroes everywhere except for the diagonal, which is filled with and for the superdiagonal, which is composed of ones.

Analytic function of a matrix

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