A Note About Measures, Jacobians and Moore–Penrose Inverse

Abstract

Some general problems of Jacobian computations in non-full rank matrices are revised in this work. We prove that the Jacobian of the Moore–Penrose inverse derived via matrix differential calculus is incorrect. In addition, the Jacobian in the full rank case is derived under the simple and old theory of the exterior product.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Billingsley, P.: Probability and Measure, 2nd edn. Wiley, New York (1986)

    Google Scholar 

  2. 2.

    Bodnar, T., Okhrin, Y.: Properties of the singular, inverse and generalized inverse partitioned Wishart distributions. J. Multivar. Anal. 99, 2389–2405 (2008)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Cadet, A.: Polar coordinates in \({\mathbf{R}}^{np}\); application to the computation of the Wishart and beta laws. Sankhyā A 58, 101–113 (1996)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Campbell, S.L., Meyer Jr., C.D.: Generalized Inverses of Linear Transformations. Pitman, London (2009)

    Google Scholar 

  5. 5.

    Díaz-García, J.A.: A note about measures and Jacobians of random matrices. J. Multivar. Anal. 98, 960–969 (2007)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Díaz-García, J.A., González-Farías, G.: Singular random matrix decompositions: Jacobians. J. Multivar. Anal. 93(2), 196–212 (2005)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Díaz-García, J.A., Gutiérrez-Jáimez, R.: Functions of singular random matrices with applications. Test 14, 475–487 (2005)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Díaz-García, J.A., Gutiérrez-Jáimez, R.: Distribution of the generalised inverse of a random matrix and its applications. J. Stat. Plan. Inference 136, 183–192 (2006)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Díaz-García, J.A., Gutiérrez-Jáimez, R.: Proof of the conjectures of H. Uhlig on the singular multivariate beta and the Jacobian of a certain matrix transformation. Ann. Stat. 25, 2018–2023 (1997)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Díaz-García, J.A., Gutiérrez-Jáimez, R., Mardia, K.V.: Wishart and Pseudo-Wishart distributions and some applications to shape theory. J. Multivar. Anal. 63, 73–87 (1997)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Evans, L.C., Garyepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press Inc., Boca Raton (1992)

    Google Scholar 

  12. 12.

    Golub, G.H., Pereyra, V.: The differentation of pseudo inverses and nonlinear least squares problems whose variables separate. SIAM J. Numer. Anal. 10(2), 413–432 (1997)

    Article  Google Scholar 

  13. 13.

    Gorecki, T., Luczak, M.: Linear discriminant analysis with a generalization of the Moore–Penrose pseudoinverse. Int. J. Appl. Math. Comput. 23(2), 463–471 (2013)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Graybill, F.A.: Theory and Application of the Linear Model. Wadsworth & Brooks/Cole, Pacific Grove (1976)

    Google Scholar 

  15. 15.

    Herz, C.S.: Bessel functions of matrix argument. Ann. Math. 61, 474–523 (1955)

    MathSciNet  Article  Google Scholar 

  16. 16.

    James, A.T.: Normal multivariate analysis and the orthogonal group. Ann. Math. Stat. 25, 40–75 (1954)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Khatri, C.G.: Some results for the singular normal multivariate regression models. Sankhyā A 30, 267–280 (1968)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Lv, X., Xiao, L., Tan, Z., Zhi, Y., Yuan, J.: Improved gradient neural networks for solving Moore–Penrose inverse of full-rank matrix. Neural Process. Lett. 50(2), 1993–2005 (2019)

    Article  Google Scholar 

  19. 19.

    Magnus, J.R.: Linear Structures. Charles Griffin & Company Ltd, London (1988)

    Google Scholar 

  20. 20.

    Mathai, A.M.: Jacobian of Matrix Transformations and Functions of Matrix Argument. World Scinentific, Singapore (1997)

    Google Scholar 

  21. 21.

    Magnus, J.R., Neudecker, H.: Matrix Differential Calculus with Application in Statistics and Econometrics, 3rd edn. Wiley, Chichester (2007)

    Google Scholar 

  22. 22.

    Muirhead, R.J.: Aspects of Multivariated Statistical Theory. Wiley, New York (2005)

    Google Scholar 

  23. 23.

    Neudecker, H., Shuangzhe, L.: The density of the Moore–Penrose inverse of a random matrix. Linear Algebra Appl. 237(238), 123–126 (1996)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Roy, S.N.: Some Aspects of Multivariate Analysis. Wiley, New York (1957)

    Google Scholar 

  25. 25.

    Spivak, M.: Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. Addison-Wesley Publishing Company, Reading (1965)

    Google Scholar 

  26. 26.

    Uhlig, H.: On singular Wishart and singular multivariate beta distributions. Ann. Stat. 22(1), 395–405 (1994)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Zhang, Y.: The exact distribution of the Mooore–Penrose inverse of \(X\) with a density. In: Krishnaiah, P.R. (ed.) Multivariate Analysis VI, pp. 633–635. Elsevier Science, Amsterdam (1985)

    Google Scholar 

Download references

Acknowledgements

The authors wish to thank the Editor and the anonymous reviewers for their constructive comments on the preliminary version of this paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to José Antonio Díaz-García.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Mohammad S. Moslehian.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Díaz-García, J.A., Caro-Lopera, F.J. A Note About Measures, Jacobians and Moore–Penrose Inverse. Bull. Iran. Math. Soc. 47, 55–62 (2021). https://doi.org/10.1007/s41980-020-00365-x

Download citation

Keywords

  • Jacobian
  • Matrix differentiation
  • Hausdorff measure
  • Lebesgue measure
  • Generalised inverse

Mathematics Subject Classification

  • 15A23
  • 15A09
  • 14R15
  • 60E05