Linear vector space derivation of new expressions for the pseudo inverse of rectangular matrices

Abstract

In this paper, a family of simple formulas for the calculation of the pseudo inverse of a rectangular matrix of less than maximum rank is derived using linear vector space methods. The principal result is that the pseudo inverse A+ of a matrix A can be calculated as A+ = Q(PTAQ)?1PT, where P and Q are rectangular matrices whose r columns are vectors that form a basis for the spaces spanned by the columns and rows, respectively, of matrix A. This leaves the user the liberty to choose the basis to take into consideration other questions such as amount of work needed and ill-conditioning of the matrix that has to be inverted. The formulas are particularized for rectangular matrices that have maximum rank and for the trivial case in which the original matrix is non-singular. Illustrative numerical examples are worked out for several choices of basis vectors and the results are compared with those provided by the program Mathematica through its function PseudoInverse[A].