Extending Pseudo Inverses for Matrices to Linear Operators in Hilbert Space

 

M. A. Murray–Lasso*¹

 

¹ Facultad de Ingeniería, Universidad Nacional Autónoma de México (UNAM). Circuito Escolar s/n, Ciudad Universitaria, Coyoacán Mexico City, Mexico 04510 *E–mail: mamurraylasso@yahoo.com

 

ABSTRACT

In this paper formulas derived by the author for calculating the pseudo inverse of any matrix are generalized to linear operators in Hilbert space. The pseudo inverse is seldom required unless there are many right side vectors, which become known at differet times. The minimum square solution of functional equations is also presented for a single right–side vector. Some definitions and theorems of functional analysis are included. An application to a simple minimum energy optimal contol problem is presented in detail.

Keywords: pseudo inverse operators, minimum norm optimization, linear operators, Hilbert space, discretization.

 

RESUMEN

En este artículo se generalizan fórmulas, deducidas por el autor, para el cálculo de la seudo inversa de cualquier matriz, a operadores lineales en espacios de Hilbert. La seudo inversa raramente se necesita a menos que muchos vectores del lado derecho se presenten en diferentes tiempos. La solución de mínimos cuadrados de ecuaciones funcionales se presenta también. Se presenta en detalle una aplicación a la solución de un problema sencillo de control óptimo de mínima energía.

 

DESCARGAR ARTÍCULO EN FORMATO PDF

 

References

[1] Murray – Lasso, M. A. (2007): "Linear Vector Space Derivation of New Expressions for the Pseudo Inverse of Rectangular Matrices", Journal of Applied Research and Technology, Vol. 5, No. 3, pp. 150 – 159.         [ Links ]

[2] Murray–Lasso, M. A. (2008): "Alternative Methods of Calculation of a Pseudo Inverse of a Non – Full Rank Matrix". Journal of Applied Research and Technology, Vol. 6, No. 3, pp. 170 – 183.         [ Links ]

[3] Dunford, N. and Schwartz, J. T. (1957): Linear Operators, Part I: General Theory. Interscience Publishers, Inc., New York.         [ Links ]

[4] Vulikh, V. Z. (1963): Functional Analysis for Scientists and Technologists. Pergamon Press, Oxford.         [ Links ]

[5] Canavati Ayub, J. A. (1998): Introducción al Análisis Funcional. Fondo de Cultura Económica, México.         [ Links ]

[6] Lanczos, C. (1961): Linear Differential Operators. D. Van Nostrand Company Limited, London        [ Links ]

[7] Kolmogorov A. N. and Fomin S. V. (1975): Elements of the Theory of Functional Analysis. Graylock Press. Rochester, NY.         [ Links ]

[8] Gille, J. C. , et al. (1959): Feedback Control Systems, Mc Graw Hill Book Company, New York.         [ Links ]

[9] Athans, M. and Falb, P. L. (1966): Optimal Control. Mc Graw – Hill Book Company, New York.         [ Links ]

[10] Wolfram, S. (1991): Mathematica. Addison – Wesley Publishing Company, Reading, MA.         [ Links ]

[11] Luenberger, D. G. (1969): Optimization by Vector Space Methods. John Wiley & Sons, Inc., New York.         [ Links ]

[12] Forsythe, G. E. and C. B. Moller (1967): Computer Solution of Linear Algebraic Systems. Prentice_hall, Inc. Englewood Cliffs, NJ.         [ Links ]

[13] Noble, B. (1969): Applied Linear Algebra. Prentice–Hall, Inc. Englewood Cliffs, NJ.         [ Links ]

[14] Dahlquist G. and A. Björck (1974): Numerical Methods. Prentice–Hall, Inc. Englewood Cliffs, NJ.         [ Links ]