A Note About Measures, Jacobians and Moore–Penrose Inverse


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.

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

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Correspondence to José Antonio Díaz-García.

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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

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

Mathematics Subject Classification

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