On the ``Reverse Order Law'' Related to the Generalized Inverse of Matrix Products

ABSTRACT

The “reverse order law” related to ordinary inverses of matrix products, i.e., (AB)-1 = B-1A-1, is generally not transferable to the generalized inverse. There are, however, applications in which the reverse order law related to the generalized inverse reveals interesting properties in certain classes of matrices. In this paper, some necessary and sufficient conditions for the reverse order property to hold are given.

REFERENCES

• 1: BEN-ISRAEL, A. An iterative method for computing the generMized inverse of an arbitrary matrix. Math. Comp. 19 (1965), 452-455.
• 2: _D CHARNES, A. Contributions to the theory of generalized inverses. J. SIAM 11 (1963), 667-699.
• 3: ---- AND WEnSN, S.J. A least squares method for computing the generalized inverse of an arbitrary matrix. ONR Res. Mere. 61, Northwestern University, The Technological Inst., Evanston, Ill., 1962.
• 4: BooT, J. C. G. The computation of the generalized inverse of singular or rectangular matrices. Am. Math. Month. 70 (1963), 302,-303.
• 5: CHIPMAN, J. S., Am) RAO, M.M. Projections, generalized inverses, and quadratic forms. J. Math. Annal. Appl. 9 (1964), 1-11.
• 6: CLINE, R. E. Note on the generalized inverse of the product of matrices. SIAM Rev. 6 (1964), 57-58.
• 7: Representations for the generalized inverse of a partitioned matrix. J. SIAM 12 (1964), 588-600.
• 8: Representations for the generalized inverse of sums of matMces. J. SIAM {B}, 2 (1965), 99-114.
• 9: DESOER, C. A., AND WHALEN, B.H. A. note on pseudoinverses. J. SIAM 1I (1963), 442-447.
• 10: ERDELYI, I. On partial isometries in finite dimensional Euclidean spaces. J. SIAM (in press).
• 11: GEWLIa, T. N.E. The pseudoinverse of a rectangular or singular matrix and its application to the solution of systems of linear equations. SIAM Rev. 1 (1959), 38--43.
• 12: ----. Some applications of the pseudoinverse of a matrix. SIAM Rein Z (1960), 15-22.
• 13: HESTENES, M. R. Inversion of matrices by biorthogonalization and related results. J. SIAM 6 (1958); 51-90.
• 14: MooRE, E.H. General Analysis, Part I. Mem. Am. Phil. Soc. i, 1935.
• 15: PENROSB, R. A generalized inverse for matrices. Proc. Camb. Phil. Soc. 51 (1955), 406-413.
• 16: PmcE, C.M. The matrix pseudoinverse and minimal variance estimates. SIAM Rev. 6 (1964), 115---120.
• 17: L. Duane Pyle, Generalized Inverse Computations Using the Gradient Projection Method, Journal of the ACM (JACM), v.11 n.4, p.422-428, Oct. 1964 [doi>10.1145/321239.321243]
• 18: RosEn, J. B. Minimum and basic solutions to singular systems. 3. SIAM 12 (1964), 156-162.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.3 Numerical Linear Algebra
          Subjects: Matrix inversion

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.1 Numerical Algorithms and Problems
          Subjects: Computations on matrices

General Terms:
Theory, Verification