Generalized Inverses: Theory, Applications and Computations

Tampere Inter University Graduate School (TISE)

Tampere University of Technology (TUT) and University of Tampere (UTA)

September 2008

Lecturer: Adi Ben-Israel, Rutgers University, NJ, USA         


Course description: Generalized inverses of matrices have important applications in statistics, engineering and many areas of applied mathematics, as well as a satisfactory theory. This short course is an introduction to the theory, and the main application and numerical algorithms of generalize inverses.


1.     The Penrose equations, and the Moore-Penrose inverse.

2.     Solving linear equations. Geometry.

3.     Applications to least squares and minimum norm solutions.

4.     Partitioned matrices and determinants.

5.     The matrix volume and applications.

6.     The group inverse.

7.     The Drazin inverse.

8.     A spectral theory for rectangular matrices. The SVDecomoposition.

9.     Applications in Statistics.

10.                        Miscellaneous applications

Prerequisites: Working knowledge of linear algebra, as can be glanced from this review

Course work, grades, etc.: Students will have a choice between an exam or a (mini)project of an applied or numerical nature, in the student's area of interest.

Textbook: None required, although I'll be reading from  Generalized Inverses: Theory and Applications

Schedule (tentative):





Topics (from Syllabus)

Wednesday, Sept 17




1, 2

Thursday, Sept 18




3, 4

Friday, Sept 19




5, 6

Wednesday, Sept 24




7, 8

Thursday, Sept 25




9, 10



·       Lecture 1: Preliminaries

·       Lecture 2: Generalized Inverses

·       Lecture 3: Least Squares

·       Lecture 4: Partitioned Matrices and Determinants

·       Lecture 5: Matrix Volume

·       Illustrations with Maple (written for Maple12, but should be compatible with older versions of Maple)

·       Lecture 6: The Group Inverse

·       Lecture 7: The Drazin Inverse

·       Lecture 8: The SVDecompoition

·       Lecture 9: Applications to Statistics

·       Lecture 10: Miscellaneous Applications

Reading list (to be updated):

1.     A. B-I, Review of linear algebra (given in chapter 0 of this file, that includes also some front matter from the above mentioned text)

2.     A. B-I, Bibliography of generalized inverses (2002)

3.     A. B-I, On error bounds for generalized inverses, SIAM J. Numer. Anal. 3(1966), 585-592

4.     A. B-I, A volume associated with m x n matrices, Lin. Algeb. and its Appl. 167(1992), 87-111

5.     A. B-I, The change of variables formula using matrix volume, SIAM Journal on Matrix Analysis 21(1999), 300-312

6.     A. B-I, An application of the matrix volume in probability, Lin. Algeb. and its Appl. 321(2001), 9-25

7.     A. B-I, The Moore of the Moore-Penrose inverse, Electron. J. Lin. Algeb. 9(2002), 150-157

8.     J.W. Blattner, Bordered matrices, J. Soc. Indust. Appl. Math. 10(1962), 528-533

9.     F. Burns, D. Carlson, E. Haynsworth and T. Markham, Generalized inverse formulas using the Schur complement, SIAM J. Appl. Math. 26(1974), 254-259

10.                     S.L. Campbell, The Drazin inverse of an infinite matrix, SIAM J. Appl. Math. 31(1976), 492-503

11.                     S.L. Campbell, Index two linear time-varying singular system of differential equations, SIAM J. Alg. Disc. Meth. 4(1983), 237-243

12.                     S.L. Campbell, C.D. Meyer and N.J. Rose, Applications of the Drazin inverse to linear systems of differential equations with singular constant coefficients, SIAM J. Appl. Math. 31(1976), 411-425

13.                     D. Constales, A closed from formula for the Moore-Penrose inverse of a complex matrix of given rank, Acta Math. Hungar. 80(1998), 83-88

14.                     M.P. Drazin, Pseudo-inverses in associative rings and semigroups, Amer. Math. Monthly 65(1958), 506-514

15.                     A. Galantai and Cs. J. Hegedus, Jordan’s principal angles in complex vector spaces, Numer. Lin. Algeb. Appl. 13(2006), 589-598

16.                     T.N.E. Greville, Some applications of the pseudoinverse of a matrix, SIAM Rev. 2(1960), 15-22

17.                     P. Kunkel and V. Mehrmann, Generalized inverses of differential-algebraic operators, SIAM J. Matrix Anal. Appl. 17(1996), 426-442

18.                     Y. Levin and A. B-I, A Newton method for systems of m equations in n variables, Nonlinear Analysis 47 (2001), 1961-1971

19.                     C.D. Meyer, The role of the group generalized inverse in the theory of finite Markov chains, SIAM Rev. 17(1975), 443-464

20.                     J. Miao and A. B-I, On principal angles between subspaces in R^n, Lin. Algeb. Appl. 171(1992), 81-98

21.                     J. Miao and A. B-I, On l_p approximate solutions of linear equations, Lin. Algeb. Appl. 199(1994), 305-327

22.                     W.T. Reid, Generalized Green’s matrices for two-point boundary problems, SIAM J. Appl. Math. 15(1967), 856-870

23.                     A. Rieder and T. Schuster, The approximate inverse in action with an application to computerized tomography, SIAM J. Numer. Anal. 37(2000), 1909-1929

24.                     C.F. Van Loan, Generalizing the SVD, SIAM J. Numer. Anal. 13(1976), 76-83