Matrix Functions of Exponential Order

Both the theoretical and practical investigations of various dynamical systems need to extend the definitions of various functions defined on the real axis to the set of matrices. To this end one uses mainly three methods which are based on 1) the Jordan canonical forms, 2) the polynomial interpolation, and 3) the Cauchy integral formula. All these methods give the same result, say g(A), when they are applicable to given function g(t) and matrix A. But, unfortunately, each of them puts certain restrictions on g(t) and/or A, and needs tedious computations to find explicit exact expressions when the eigen-values of A are not simple. The aim of the present paper is to give an alternate method which is more logical, simple and applicable to all functions (continuous or discontinuous) of exponential order. It is based on the two-sided Laplace transform and analytical continuation concepts, and gives the result as a linear combination of certain n matrices determined only through A. Here n stands for the order of A. The coefficients taking place in the combination in question are given through the analytical continuation of g(t) (and its derivatives if A has multiple eigen-values) to the set of eigen-values of A (numerical computation of inverse transforms is not needed). Some illustrative examples show the effectiveness of the method.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

M. Idemen, "Matrix Functions of Exponential Order," Applied Mathematics, Vol. 4 No. 9, 2013, pp. 1260-1268. doi: 10.4236/am.2013.49170.

  A. Cayley, “A Memoir on the Theory of Matrices,” Philosophical Transactions of the Royal Society of London, Vol. 148, 1858, pp. 17-37. doi:10.1098/rstl.1858.0002  N. J. Higham, “Functions of Matrices: Theory and Com putation,” Society for Industrial and Applied Mathe matica (SIAM), Philadelphia, 2008.  N. I. Mushkhelishvili, “Singular Integral Equations,” P. Noordhoff Ltd., Holland, 1958.  E. C. Titchmarsh, “Introduction to the Theory of Fourier Integrals,” Chelsea Publishing Company, 1986, Chapter 1.3. 