Artículo Original
Publicado 2018-06-18
Palabras clave
- Función matricial,
- forma canónica de Jordan,
- derivada de Fréchet
Cómo citar
Marmolejo, M. A. (2018). Forma de Jordan de la derivada de Fréchet de funciones matriciales. Revista Integración, Temas De matemáticas, 36(1), 1–19. https://doi.org/10.18273/revint.v36n1-2018001
Resumen
En este artículo se presenta una fórmula para evaluar funciones matriciales f : A ⊂ C 2×2 → C 2×2, en términos de dos funciones escalares que sólo dependen de la traza y el determinante de X ∈ C 2×2 . Se explota el conocimiento de las derivadas de Fréchet de las funciones traza y determinante para determinar la derivada de Fréchet de f(·). Como resultado central, se da la forma canónica de Jordan de la derivada de Fréchet Df(X) : C 2×2 → C 2×2.
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Referencias
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[2] Deadman E. and Relton S.D., “Taylor’s theorem for matrix functions with applications to condition number estimation”, Linear Algebra Appl. 504 (2016), 354–371.
[3] Golub G. and Van Loan C., Matrix Computations, The Johns Hopkins University Press, Baltimore, 1996.
[4] Higham N.J., Functions of Matrices: Theory and Computation, Siam, Philadelphia, 2008.
[5] Higham and Al-Mohy A.H., “Computing matrix functios”, Acta Numer. 19 (2010), 159–208.
[6] Higham N.J. and Lijing L., “Matrix Functions: A short Course”, en Ser. Contemp. Appl. Math. 19, Higher Ed. Press, Beijing, (2015), 1–27.
[7] Horn R.A. and Johnson C.R., Topics in Matrix Analysis, Cambridge University Press, New York, 1991.
[8] Kandolf P. and Relton S.D., “A block Krylov method to compute the action of the Fréchet derivative of a matrix function on a vector with applications to condition number estimation”, Siam J. Sci. Comp. 39 (2017), No. 4, A1416–A1434.
[9] Najfeld I. and Havel T.F., “Derivatives of the matrix exponential and their computation”, Adv. in Appl. Math. 16 (1995), No. 3, 321–375.
[10] Rinehart R.F., “The Equivalence of Definitions of a Matric Function”, Amer. Math. Monthly 62 (1955), No. 6, 395-414.
[11] Stickel E., “On the Fréchet Derivative of Matrix Functions”, Linear Algebra Appl. 91 (1987), 83–88.