Analogs of the adjoint matrix for generalized inverses and corresponding Cramer rules

In this article, we introduce determinantal representations of the Moore–Penrose inverse and the Drazin inverse which are based on analogs of the classical adjoint matrix. Using the obtained analogs of the adjoint matrix, we get Cramer rules for the least squares solution and for the Drazin inverse solution of singular linear systems. Finally, determinantal expressions for A + A, A A +, and A D A are presented.

[1]  Guorong Wang,et al.  A Cramer rule for minimum-norm (T) least-squares (S) solution of inconsistent linear equations , 1986 .

[2]  M. Drazin.  Pseudo-Inverses in Associative Rings and Semigroups , 1958 .

[3]  Hebing Wu,et al.  ADDITIONAL RESULTS ON INDEX SPLITTINGS FOR DRAZIN INVERSE SOLUTIONS OF SINGULAR LINEAR SYSTEMS , 2001 .

[4]  Predrag S. Stanimirović,et al.  Full-rank and determinantal representation of the Drazin inverse , 2000 .

[5]  Adi Ben-Israel.  Generalized inverses of matrices: a perspective of the work of Penrose , 1986, Mathematical Proceedings of the Cambridge Philosophical Society.

[6]  Adi Ben-Israel,et al.  A Cramer rule for least-norm solutions of consistent linear equations , 1982 .

[7]  Guo-rong Wang.  A Cramer rule for finding the solution of a class of singular equations , 1989 .

[8]  Hans Joachim Werner,et al.  On extensions of Cramer's rule for solutions of restricted linear systems 1 , 1984 .

[9]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[10]  R. Bapat,et al.  Generalized inverses over integral domains , 1990 .

[11]  R. Chan,et al.  Preconditioners for non‐Hermitian Toeplitz systems , 2001 .

[12]  J. Meyer,et al.  Limits and the Index of a Square Matrix , 1974 .

[13]  Yong-Lin Chen.  A cramer rule for solution of the general restricted linear equation , 1993 .

[14]  Jun Ji.  Explicit expressions of the generalized inverses and condensed Cramer rules , 2005 .