Algebraic Aspects of the Generalized Inverse of a Rectangular Matrix

Publisher Summary This chapter discusses the algebraic aspects of the generalized inverse of a rectangular matrix. When Penrose rediscovered the notion of the generalized inverse of a rectangular matrix, his point of view and proofs were purely algebraic. It soon became clear that Penrose's axioms were equivalent to the earlier definition of Moore, which was expressed in a rather different language. The technique used by subsequent authors gradually shifted from the algebraic toward Hubert-space geometric. The generalized inverse is an algebraic notion and it is of some interest to ask what the structure of a Hilbert space has to do with it. Positivity can be imposed on the theory ex post facto after the generalized inverse has been defined over an arbitrary field. The chapter develops the theory of the generalized inverse ab initio in a strictly linear- algebraic fashion. The generalized inverse is not an invariant notion. It is not a basis-free property of a linear transformation but a gadget constructed from a rectangular array of numbers. Neither a geometric nor an abstract-algebraic formalism is really appropriate; the most natural approach, very similar to Penrose's, is to write all formulas in terms of matrices obtained directly from the given rectangular matrix A .