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Generalization of the Cramer’s Rule and the Minors of the Generalized Inverses

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Generalized Inverses: Theory and Computations

Part of the book series: Developments in Mathematics ((DEVM,volume 53))

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Abstract

It is well known that the Cramer’s rule for the solution \(\mathbf {x}\) of a nonsingular equation

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References

  1. S.M. Robinson, A short proof of Cramer’s rule. Math. Mag. 43, 94–95 (1970). Reprinted in Selected Papers on Algebra (S. Montgomery et al. eds.) Math. Assoc. Amer. 313–314 (1977)

    Article  MathSciNet  Google Scholar 

  2. A. Ben-Israel, A Cramer rule for least-norm solution of consistent linear equations. Linear Algebra Appl. 43, 223–226 (1982)

    Article  MathSciNet  Google Scholar 

  3. Y. Chen, A Cramer rule for solution of the general restricted linear equation. Linear Multilinear Algebra 34, 177–186 (1993)

    Article  MathSciNet  Google Scholar 

  4. Y. Chen, Expressions and determinantal formulas for solution of a restricted matrix equation. Acta Math. Appl. Sinica 18(1), 65–73 (1995)

    MathSciNet  MATH  Google Scholar 

  5. J. Ji, An alternative limit expression of Drazin inverse and its applications. Appl. Math. Comput. 61, 151–156 (1994)

    MathSciNet  MATH  Google Scholar 

  6. W. Sun, Cramer rule for weighted system. J. Nanjing Univ. 3, 117–121 (1986). in Chinese

    MathSciNet  MATH  Google Scholar 

  7. G.C. Verghes, A "Cramer rule" for least-norm least-square-error solution of inconsistent linear equations. Linear Algebra Appl. 48, 315–316 (1982)

    Article  MathSciNet  Google Scholar 

  8. G. Wang, A Cramer rule for minimum-norm\((T)\) least-squares\((S)\) solution of inconsistent equations. Linear Algebra Appl. 74, 213–218 (1986)

    Article  MathSciNet  Google Scholar 

  9. G. Wang, A Cramer rule for finding the solution of a class of singular equations. Linear Algebra Appl. 116, 27–34 (1989)

    Article  MathSciNet  Google Scholar 

  10. G. Wang, On extensions of cramer rule. J. Shanghai Normal Univ. 21, 1–7 (1992)

    MathSciNet  Google Scholar 

  11. G. Wang, Z. Wang, More condensed determinant representation for the best approximate solution of the matrix equation \(AXH=K\). J. Shanghai Normal Univ. 19, 27–31 (1990). in Chinese

    Google Scholar 

  12. H.J. Werner, When is \(B^-A^-\) a generalized inverse of \(AB\)? Linear Algebra Appl. 210, 255–263 (1994)

    Article  MathSciNet  Google Scholar 

  13. G. Wang, On the singularity of a certain bordered matrix and its applications to the computation of generalized inverses. J. Shanghai Normal Univ. 18, 7–14 (1989). in Chinese

    Google Scholar 

  14. H.J. Werner, On extensions of Cramer’s rule for solutions of restricted linear systems. Linear Multilinear Algebra 15, 319–330 (1984)

    Article  MathSciNet  Google Scholar 

  15. Y. Chen, B. Zhou, On g-inverses and the nonsingularity of a bordered matrix . Linear Algebra Appl. 133, 133–151 (1990)

    Google Scholar 

  16. Y. Wei, A characterization and representation of the generalized inverse \(A_{T, S}^{(2)}\) and its applications. Linear Algebra Appl. 280, 87–96 (1998)

    Article  MathSciNet  Google Scholar 

  17. R. Penrose, On best approximate solution of linear matrix equations. Proc. Cambridge Philos. Soc. 52, 17–19 (1956)

    Article  MathSciNet  Google Scholar 

  18. J. Ji, The cramer rule for the best approximate solution of the matrix equation \(AXB=D\). J. Shanghai Normal Univ. 14, 19–23 (1985). in Chinese

    MATH  Google Scholar 

  19. M. Marcus, M. Minc, A Survey of Matrix Theory and Matrix Inequalities (Mass, Allyn and Bacon, Boston, 1964)

    MATH  Google Scholar 

  20. G. Wang, The Generalized Inverses of Matrices and Operators (Science Press, Beijing, 1994). in Chinese

    Google Scholar 

  21. G. Wang, Weighted Moore-Penrose, Drazin, and group inverses of the Kronecker product \(A \otimes B\) and some applications. Linear Algebra Appl. 250, 39–50 (1997)

    Article  MathSciNet  Google Scholar 

  22. J. Ji, Explicit expressions of the generalized inverses and condensed cramer rules. Linear Algebra Appl. 404, 183–192 (2005)

    Article  MathSciNet  Google Scholar 

  23. Y. Wei, A characterization for the W-weighted Drazin inverse and Cramer rule for W-weighted Drazin inverse solution. Appl. Math. Comput. 125, 303–310 (2002)

    MathSciNet  MATH  Google Scholar 

  24. R.A. Brualdi, H. Schneider, Determinantal: gauss, schur, cauchy, sylvester, kronecker, jacobi, binet, laplace, muir and cayley. Linear Algebra Appl. 52, 769–791 (1983)

    Article  MathSciNet  Google Scholar 

  25. A. Ben-Israel, Minors of the Moore-Penrose inverse. Linear Algebra Appl. 195, 191–207 (1993)

    Article  MathSciNet  Google Scholar 

  26. A. Ben-Israel, A volume associated with \(m \times n\) matrices. Linear Algebra Appl. 167, 87–111 (1992)

    Article  MathSciNet  Google Scholar 

  27. G. Wang, J. Sun, J. Gao. Minors of the weighted Moore-Penrose inverse. Numer. Math., J. Chinese Univ. 19(4), 343–348 (1997). in Chinese

    Google Scholar 

  28. R.A. Horn, C.R. Johnson, Matrix Analysis, 2nd edn. (Cambridge University Press, Cambridge, 2012)

    Book  Google Scholar 

  29. J. Miao, Reflexive generalized inverses and their minors. Linear Multilinear Algebra 35, 153–163 (1993)

    Article  MathSciNet  Google Scholar 

  30. G. Wang, J. Gao, J. Sun, Minors of Drazin inverses. J. Shanghai Normal Univ. 28, 12–15 (1998). in Chinese

    Google Scholar 

  31. G. Wang, J. Gao, Minors of the generalized inverse \(A_{T, S}^{(2)}\). Math. Numer. Sinica 23(4), 439–446 (2001). in Chinese

    Google Scholar 

  32. Y. Yu, The generalized inverse \(A_{T,S}^{(2)}\) on an associative ring: theory and computation. Ph.D. thesis, Shanghai Normal University, China, 2006. in Chinese

    Google Scholar 

  33. Y. Chen, Finite algorithm for the \((2)\)-generalized inverse \(A_{T, S}^{(2)}\). Linear Multilinear Algebra 40, 61–68 (1995)

    Article  MathSciNet  Google Scholar 

  34. R.B. Bapat, A. Ben-Israel, Singular values and maximum rank minors of generalized inverses. Linear Multilinear Algebra 40, 153–161 (1995)

    Article  MathSciNet  Google Scholar 

  35. R.E. Hartwig, Singular value decomposition and the Moore-Penrose inverse of bordered matices. SIAM J. Appl. Math. 31, 31–41 (1976)

    Article  MathSciNet  Google Scholar 

  36. R.E. Hartwig, Block generalized inverses. Arch Rational Mech. Anal. 61, 197–251 (1976)

    Article  MathSciNet  Google Scholar 

  37. J. Miao, Representations for the weighted Moore-Penrose inverse of a partitioned matrix. J. Comput. Math. 7, 320–323 (1989)

    MathSciNet  Google Scholar 

  38. J. Miao, Some results for computing the Drazin inverse of a partitioned matrix. J. Shanghai Normal Univ. 18, 25–31 (1989). in Chinese

    Google Scholar 

  39. M. Wei, W. Guo, On g-inverses of a bordered matrix: revisited. Linear Algebra Appl. 347, 189–204 (2002)

    Article  MathSciNet  Google Scholar 

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Correspondence to Guorong Wang .

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Wang, G., Wei, Y., Qiao, S. (2018). Generalization of the Cramer’s Rule and the Minors of the Generalized Inverses . In: Generalized Inverses: Theory and Computations. Developments in Mathematics, vol 53. Springer, Singapore. https://doi.org/10.1007/978-981-13-0146-9_3

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