Title: Appendix C: One-Sided and Generalized Inverses
Abstract: In the main text we use some well-known facts about one-sided and generalized inverses. These facts, in an appropriate form, are presented here with their proofs.Let A be an m × n matrix with complex entries. A is called left invertible (resp. right invertible) if there exists an n × m complex matrix AI such that AIA=I (resp. AAI=I ). In this case AI is called a left (resp. right) inverse of A.The notion of one-sided invertibility is a generalization of the notion of invertibility (nonsingularity) of square matrices. If A is a square matrix and det A≠0 , then A−1 is the unique left and right inverse of A.One-sided invertibility is easily characterized in other ways. Thus, in the case of left invertibility, the following statements are equivalent;(i) the m × n matrix A is left invertible;(ii) the columns of A are linearly independent in Ȼm (in particular, m≥n );(iii) KerA={0} , where A is considered as a linear transformation from Ȼn to Ȼm .Let us check this assertion. Assume A is left invertible, and assume that Ax=0 for some x∈Ȼn . Let AI be a left inverse of A. Then x=AIAx=0 i.e., KerA={0} , so (i) ⇒ (iii) follows.
Publication Year: 2009
Publication Date: 2009-01-01
Language: en
Type: other
Indexed In: ['crossref']
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