An effective recursive formula for the Frobenius covariants in matrix functions

F. Schäfer 1
  • 1 Institute for Mathematics, University of Klagenfurt, 9020 , Klagenfurt, Austria

Abstract

For theoretical studies, it is helpful to have an explicit expression for a matrix function. Several methods have been used to determine the required Frobenius covariants. This paper presents a recursive formula that calculates these covariants effectively. The new aspect of this method is the simple determination of the occurring coefficients in the covariants. The advantage is shown by several examples for the matrix exponential in comparision with Mathematica. The calculations are performed exactly.

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  • [1] M. Ahmadvand and A. Sadeghi. New formulation for the matrix roots with multiple eigenvalues. International Journal of Modern Sciences and Engineering Technology (IJMSET), 1(4):30-37, 2014.

  • [2] W. Balser. Lineare Algebra 2. http://www.mathematik.uni-ulm.de/m5/balser/Skripten/LA2.pdf, 2008. [Lecture notes SS2008 p. 22-33, (reading at December 2015)].

  • [3] R. Ben Taher and M. Rachidi. Explicit formulas for the constituent matrices. Application to the matrix functions. Special Matrices, 3(1), 2015.

  • [4] F. Chang. A direct approach to the constituent matrices of an arbitrary matrix with multiple eigenvalues. Proceedings of the IEEE, 65(10):1509-1510, 1977.

  • [5] W. Forst and D. Hoffmann. Gewöhnliche Differentialgleichungen. Springer, 2005.

  • [6] F. Gantmacher. Matrizentheorie. Springer, 1986.

  • [7] N. J. Higham. Functions of Matrices, Theory and Computation. Siam, 2008.

  • [8] R. Horn and C. Johnson. Topics in matrix analysis. Cambridge University Press, 1991.

  • [9] P.-F. Hsieh and Y. Sibuya. Basic theory of ordinary differential equations. Springer Science & Business Media, 2012.

  • [10] C. Moler and C. Van Loan. Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Review, 45(1):3-49, 2003.

  • [11] R. Rinehart. The equivalence of definitions of a matric function. The American Mathematical Monthly, 62(6):395-414, 1955.

  • [12] A. Sadeghi, A. I. M. Ismail, and A. Ahmad. Computing the pth roots of a matrix with repeated eigenvalues. Applied Mathematical Sciences, 5(53):2645-2661, 2011.

  • [13] H. Schwerdtfeger. Beiträge zum Matricen-Kalkül und zur Theorie der Gruppenmatrix. PhD thesis, University of Bonn, Germany, 1935.

  • [14] H. Schwerdtfeger. Über mehrdeutige Matrixfunktionen. Compositio Mathematica, 3:380-390, 1936.

  • [15] A. Spitzbart. A generalization of Hermite’s interpolation formula. The American Mathematical Monthly, 67(1):42-46, 1960.

  • [16] L. Verde-Star. Divided differences and linearly recursive sequences. Studies in Applied Mathematics, 95(4):433-456, 1995.

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