SFB 303 Discussion Paper No. B - 006

Author: Sondermann, Dieter
Title: Best Approximate Solutions to Matrix Equations under Rank Restrictions
Abstract: The paper extends Penrose's (1955) concept of best approximate solutions of matrix equations to linear equations with rank restrictions. We show that the matrix equation

under the restriction
rank (X) smaller or equal K

for arbitrary matrices A, B, C and integer K has a unique best approximate solution in the sense of Penrose (1955). This generalizes and sharpens results due to Schmidt (1907), Eckart and Young (1936), Householder and Young (1938), Fisher (1969) and Rao (1980).

Already the classical result where A , B or C are the identity matrix has numerous statistical applications in the theory of indices (see e. g. Schneeweiß (1965) and Bamberg and Spremann (1984)), factor analysis, principal component analysis, multidimensional scaling and graphical data analysis techniques. For a review of these statistical applications we refer to Gabriel (1971), Gnanadesikan (1977) and Kruskal (1978).
Creation-Date: October 1985 
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