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Notice on the Chipman generalization of the matrix inverse. (English) Zbl 0964.65043
The Chipman generalized inverse of an \(m\times n\) real matrix \(A\) is the unique matrix \(A^+_{MN} \) that solves the minimum norm least squares problem with respect to the norms associated to the real positive definite matrices \(N\) and \(M\) in the domain and range, respectively. If instead of positive definite one has positive semidefinite matrices, uniqueness is no longer valid, and it is possible to exhibit a family of generalized inverses of \(A\) that still solve the minimum norm least squares problem. The scope of the paper is to give a new proof showing this.

MSC:
65F20 Numerical solutions to overdetermined systems, pseudoinverses
15A09 Theory of matrix inversion and generalized inverses
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References:
[1] Chipman J. S.: On least squares with insufficient observations. J. Amer. Statist. Assoc. 59 (1964), 1078-1111. · Zbl 0144.42401 · doi:10.2307/2282625
[2] Rao C. R., Mitra K. S.: Generalized Inverse of Matrices and Its Application. J. Wiley, New York 1971. · Zbl 0236.15004
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