By Rotman J.J.

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P zutrifft, ist (65) Damit haben wit gezeigt, daB die Gleichung (62) immer genau eine beste Naherungslasung hat, die durch (65) definiert ist. 1st Y = E, die Einheitsmatrix m-ter Ordnung, so erhalten wir XO = A+. Folglich ist die pseudoinverse Matnx A+ die beste NaherungslOsung (nach der Methode der kleinsten Quadratsumme) der M atrizengleichung AX=E. Diese Eigenschaft der pseudoinversen Matrix A + kann auch als ihre Definition genommen werden. 5. Die Grevillesche Methode zur sukzessiven Bestimmung der pseudoinversen Matrix.

N) (23) k=l die entsprechende lineare Transformation. , - " ' " Ik 17k '-IAI ............................. aliI ••• all. , n) . " (24) 34 1. Matrizen und Matrizenoperationen Damit haben wir die zu (23) inverse Transformation erhalten. Die Koeffizientenmatrix dieser Transformation, A-I = lIa~kl)II~, hei13t die zu A inver8e Matrix (oder Matrix). ~, k = 1, 2 , ... ,n. ) (-I) a"k =Aki - • rez~proke IAI (25) Dabei sind die Aki die algebraischen Komplemente (Adjunkten) der Elemente aik in der Determinante IAI (i, k = 1,2, ...

4. Zerlegung quadratischer Matrizen in Produkte von Dreiecksmatrizen 65 3. Die GauBsche Eliminationsmethode definiert, wenn sie auf eine Matrix A = Ilaikll~ vom Rang r mit Dk =1= 0 fUr k = 1,2, ... , r angewendet wird, zwei Matrizen: die untere Dreiecksmatrix W, deren Diagonalelemente gleich 1, und die ob ere Dreiecksmatrix G, deren erste r Diagonalelemente gleich D 1, ~ -, Dl 4 ••• , - - D r- 1 sind; die Elemente der letzten n - r Zeilen der Matrix G sind gleich O. Gist die GaufJsche Form der Matrix A, W ihre Transformationsmatrix.

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