A new proof for Decell's finite algorithm for generalized inverses (Q797954)

This paper presents a new proof for \textit{H. P. Decell's} finite algorithm [SIAM REv. 7, 526-528 (1965; Zbl 0178.355)] for finding the pseudoinverse of a rectangular matrix. The main result is stated as follows. For Z a given \(m\times n\) matrix of real elements of rank r, the Moore-Penrose generalized inverse (pseudoinverse) of Z is \(Z^+=A_ r/\delta_ rZ^ T\), \(\delta_ r\neq 0\) where \(A_ r\), \(\delta_ r\) are generated recursively by \(A_{k+1}=\delta_ kI-Z^ TZA_ k, \delta_{k+1}=(1/k+1)tr(Z^ TZA_{k+1}), k=0,1,...,n\), with initial conditions \(\delta_ 0=1\), \(A_ 0=0\) and where I denotes the \(n\times n\) identity matrix. The authors develop some additional properties of Decell's algorithm and point out how it can be used in the development of algebraic properties of the pseudoinverse.