Summary
TheMoore-Penrose generalized inverse for the matrix, in the light of recent developments in control, filtering, estimation theory, and pattern recognition, has been shown to have a great deal of importance as far as applications are concerned. In the previous literature, the theory used to derive the generalized inverse seems to have obscured the fundamental properties of what the generalized inverse is. This paper gives an alternate method of deriving the generalized inverse using classical minimization theory which is familiar and has the advantage that the application of the generalized inverse is made clear as a part of the development. This derivation results in two simultaneous matrix equations which are then shown to be equivalent to the four equations used byPenrose to define the generalized inverse. These two equations make possible the direct calculation of the generalized inverse of matrix without resorting to conventional iterated computational schemes.
Zusammenfassung
Die Moore-Penrose verallgemeinerte Inverse einer Matrix ist sehr wichtig für die Anwendung aus der Sicht neuerer Entwicklungen in Regelungs-, Filter- und Schätz-Theorie und Zeichenerkennung. In der bisherigen Literatur scheint die zur Herleitung der verallgemeinerten Inversen verwendete Theorie die Grundeigenschaften der verallgemeinerten Inversen im Dunkeln gelassen zu haben. Diese Arbeit verwendet zur Herleitung der verallgemeinerten Inversen die klassische Minimisierungstheorie, welche allgemein bekannt ist und den Vorteil hat, daß die Anwendung klar als Teil der Herleitung erscheint. Die Ableitung liefert zwei simultane Matrixgleichungen, von denen gezeigt wird, daß sie äquivalent zu den vier Gleichungen sind, die Penrose für die Definition der verallgemeinerten Inversen verwendet. Diese zwei Gleichungen ermöglichen die direkte Berechnung der verallgemeinerten Inversen einer Matrix ohne zu gebräuchlichen Iterationsverfahren Zuflucht zu nehmen.
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Moore, E. H.: On the Reciprocal of the General Algebraic Matrix. Abstract. Bull. Amer. Math. Soc.26, 394–395 (1970).
Bjerhammar, A.: Rectangular Reciprocal Matrices with Special Reference to Geodetical Calculations. Bull. Géodésique1951, 188–220.
Penrose, R.: A Generalized Inverse for Matrices. Proc. Cam. Phil. Soc.51, 406–413 (1955).
Penrose, R.: On Best Approximate Solutions of Linear Matrix Equations. Proc. Cam. Phil. Soc.52, 17–19 (1956).
Zadeh andDesoer: Linear System Theory, pp. 573–574. McGraw-Hill Co. 1963.
Decell, H. P.: An Application of the Cayley-Hamilton Theory to Generalized Matrix Inversion. SIAM Review7 (1965).
Voith, R. P.: Calculation of the Generalized Inverse of a Matrix. Master of Science Thesis, University of Pittsburgh. 1968.
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Voith, R.P., Vogt, W.G. & Mickle, M.H. On the computation of the generalized inverse by classical minimization. Computing 9, 175–187 (1972). https://doi.org/10.1007/BF02246728
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DOI: https://doi.org/10.1007/BF02246728