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...

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Fecha de publicación:: 2020

Institución:: Universidad de Medellín

Repositorio:: Repositorio UDEM

Idioma:: eng

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repository_id_str
dc.title.none.fl_str_mv	A Note About Measures, Jacobians and Moore–Penrose Inverse
title	A Note About Measures, Jacobians and Moore–Penrose Inverse
spellingShingle	A Note About Measures, Jacobians and Moore–Penrose Inverse Generalised inverse Hausdorff measure Jacobian Lebesgue measure Matrix differentiation
title_short	A Note About Measures, Jacobians and Moore–Penrose Inverse
title_full	A Note About Measures, Jacobians and Moore–Penrose Inverse
title_fullStr	A Note About Measures, Jacobians and Moore–Penrose Inverse
title_full_unstemmed	A Note About Measures, Jacobians and Moore–Penrose Inverse
title_sort	A Note About Measures, Jacobians and Moore–Penrose Inverse
dc.subject.none.fl_str_mv	Generalised inverse Hausdorff measure Jacobian Lebesgue measure Matrix differentiation
topic	Generalised inverse Hausdorff measure Jacobian Lebesgue measure Matrix differentiation
description	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. © 2020, Iranian Mathematical Society.
publishDate	2020
dc.date.accessioned.none.fl_str_mv	2020-04-29T14:53:33Z
dc.date.available.none.fl_str_mv	2020-04-29T14:53:33Z
dc.date.none.fl_str_mv	2020
dc.type.eng.fl_str_mv	Article
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dc.identifier.issn.none.fl_str_mv	10186301
dc.identifier.uri.none.fl_str_mv	http://hdl.handle.net/11407/5647
dc.identifier.doi.none.fl_str_mv	10.1007/s41980-020-00365-x
identifier_str_mv	10186301 10.1007/s41980-020-00365-x
url	http://hdl.handle.net/11407/5647
dc.language.iso.none.fl_str_mv	eng
language	eng
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dc.relation.references.none.fl_str_mv	Billingsley, P., (1986) Probability and Measure, , 2, Wiley, New York Bodnar, T., Okhrin, Y., Properties of the singular, inverse and generalized inverse partitioned Wishart distributions (2008) J. Multivar. Anal., 99, pp. 2389-2405 Cadet, A., Polar coordinates in Rnp application to the computation of the Wishart and beta laws (1996) Sankhy? A, 58, pp. 101-113 Campbell, S.L., Meyer, C.D., Jr., (2009) Generalized Inverses of Linear Transformations, , Pitman, London Díaz-García, J.A., A note about measures and Jacobians of random matrices (2007) J. Multivar. Anal., 98, pp. 960-969 Díaz-García, J.A., González-Farías, G., Singular random matrix decompositions: Jacobians (2005) J. Multivar. Anal., 93 (2), pp. 196-212 Díaz-García, J.A., Gutiérrez-Jáimez, R., Functions of singular random matrices with applications (2005) Test, 14, pp. 475-487 Díaz-García, J.A., Gutiérrez-Jáimez, R., Distribution of the generalised inverse of a random matrix and its applications (2006) J. Stat. Plan. Inference, 136, pp. 183-192 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 (1997) Ann. Stat., 25, pp. 2018-2023 Díaz-García, J.A., Gutiérrez-Jáimez, R., Mardia, K.V., Wishart and Pseudo-Wishart distributions and some applications to shape theory (1997) J. Multivar. Anal., 63, pp. 73-87 Evans, L.C., Garyepy, R.F., (1992) Measure Theory and Fine Properties of Functions, , CRC Press Inc., Boca Raton Golub, G.H., Pereyra, V., The differentation of pseudo inverses and nonlinear least squares problems whose variables separate (1997) SIAM J. Numer. Anal., 10 (2), pp. 413-432 Gorecki, T., Luczak, M., Linear discriminant analysis with a generalization of the Moore Penrose pseudoinverse (2013) Int. J. Appl. Math. Comput., 23 (2), pp. 463-471 Graybill, F.A., (1976) Theory and Application of the Linear Model, , Wadsworth & Brooks/Cole, Pacific Grove Herz, C.S., Bessel functions of matrix argument (1955) Ann. Math., 61, pp. 474-523 James, A.T., Normal multivariate analysis and the orthogonal group (1954) Ann. Math. Stat., 25, pp. 40-75 Khatri, C.G., Some results for the singular normal multivariate regression models (1968) Sankhy? A, 30, pp. 267-280 Lv, X., Xiao, L., Tan, Z., Zhi, Y., Yuan, J., Improved gradient neural networks for solving Moore Penrose inverse of full-rank matrix (2019) Neural Process. Lett., 50 (2), pp. 1993-2005 Magnus, J.R., (1988) Linear Structures, , Charles Griffin & Company Ltd, London Mathai, A.M., (1997) Jacobian of Matrix Transformations and Functions of Matrix Argument, , World Scinentific, Singapore Magnus, J.R., Neudecker, H., (2007) Matrix Differential Calculus with Application in Statistics and Econometrics, , 3, Wiley, Chichester Muirhead, R.J., (2005) Aspects of Multivariated Statistical Theory, , Wiley, New York Neudecker, H., Shuangzhe, L., The density of the Moore Penrose inverse of a random matrix (1996) Linear Algebra Appl., 237 (238), pp. 123-126 Roy, S.N., (1957) Some Aspects of Multivariate Analysis, , Wiley, New York Spivak, M., (1965) Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus, , Addison-Wesley Publishing Company, Reading Uhlig, H., On singular Wishart and singular multivariate beta distributions (1994) Ann. Stat., 22 (1), pp. 395-405 Zhang, Y., The exact distribution of the Mooore Penrose inverse of X with a density (1985) Multivariate Analysis VI, pp. 633-635. , Krishnaiah PR, (ed), Elsevier Science, Amsterdam
dc.rights.coar.fl_str_mv	http://purl.org/coar/access_right/c_16ec
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dc.publisher.none.fl_str_mv	Springer
dc.publisher.program.none.fl_str_mv	Facultad de Ciencias Básicas
dc.publisher.faculty.none.fl_str_mv	Facultad de Ciencias Básicas
publisher.none.fl_str_mv	Springer
dc.source.none.fl_str_mv	Bulletin of the Iranian Mathematical Society
institution	Universidad de Medellín
repository.name.fl_str_mv	Repositorio Institucional Universidad de Medellin
repository.mail.fl_str_mv	repositorio@udem.edu.co
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spelling	20202020-04-29T14:53:33Z2020-04-29T14:53:33Z10186301http://hdl.handle.net/11407/564710.1007/s41980-020-00365-xSome 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. © 2020, Iranian Mathematical Society.engSpringerFacultad de Ciencias BásicasFacultad de Ciencias Básicashttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85079634164&doi=10.1007%2fs41980-020-00365-x&partnerID=40&md5=59e022b080e4fa77192f27d3a35b0305Billingsley, P., (1986) Probability and Measure, , 2, Wiley, New YorkBodnar, T., Okhrin, Y., Properties of the singular, inverse and generalized inverse partitioned Wishart distributions (2008) J. Multivar. Anal., 99, pp. 2389-2405Cadet, A., Polar coordinates in Rnpapplication to the computation of the Wishart and beta laws (1996) Sankhy? A, 58, pp. 101-113Campbell, S.L., Meyer, C.D., Jr., (2009) Generalized Inverses of Linear Transformations, , Pitman, LondonDíaz-García, J.A., A note about measures and Jacobians of random matrices (2007) J. Multivar. Anal., 98, pp. 960-969Díaz-García, J.A., González-Farías, G., Singular random matrix decompositions: Jacobians (2005) J. Multivar. Anal., 93 (2), pp. 196-212Díaz-García, J.A., Gutiérrez-Jáimez, R., Functions of singular random matrices with applications (2005) Test, 14, pp. 475-487Díaz-García, J.A., Gutiérrez-Jáimez, R., Distribution of the generalised inverse of a random matrix and its applications (2006) J. Stat. Plan. Inference, 136, pp. 183-192Díaz-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 (1997) Ann. Stat., 25, pp. 2018-2023Díaz-García, J.A., Gutiérrez-Jáimez, R., Mardia, K.V., Wishart and Pseudo-Wishart distributions and some applications to shape theory (1997) J. Multivar. Anal., 63, pp. 73-87Evans, L.C., Garyepy, R.F., (1992) Measure Theory and Fine Properties of Functions, , CRC Press Inc., Boca RatonGolub, G.H., Pereyra, V., The differentation of pseudo inverses and nonlinear least squares problems whose variables separate (1997) SIAM J. Numer. Anal., 10 (2), pp. 413-432Gorecki, T., Luczak, M., Linear discriminant analysis with a generalization of the Moore Penrose pseudoinverse (2013) Int. J. Appl. Math. Comput., 23 (2), pp. 463-471Graybill, F.A., (1976) Theory and Application of the Linear Model, , Wadsworth & Brooks/Cole, Pacific GroveHerz, C.S., Bessel functions of matrix argument (1955) Ann. Math., 61, pp. 474-523James, A.T., Normal multivariate analysis and the orthogonal group (1954) Ann. Math. Stat., 25, pp. 40-75Khatri, C.G., Some results for the singular normal multivariate regression models (1968) Sankhy? A, 30, pp. 267-280Lv, X., Xiao, L., Tan, Z., Zhi, Y., Yuan, J., Improved gradient neural networks for solving Moore Penrose inverse of full-rank matrix (2019) Neural Process. Lett., 50 (2), pp. 1993-2005Magnus, J.R., (1988) Linear Structures, , Charles Griffin & Company Ltd, LondonMathai, A.M., (1997) Jacobian of Matrix Transformations and Functions of Matrix Argument, , World Scinentific, SingaporeMagnus, J.R., Neudecker, H., (2007) Matrix Differential Calculus with Application in Statistics and Econometrics, , 3, Wiley, ChichesterMuirhead, R.J., (2005) Aspects of Multivariated Statistical Theory, , Wiley, New YorkNeudecker, H., Shuangzhe, L., The density of the Moore Penrose inverse of a random matrix (1996) Linear Algebra Appl., 237 (238), pp. 123-126Roy, S.N., (1957) Some Aspects of Multivariate Analysis, , Wiley, New YorkSpivak, M., (1965) Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus, , Addison-Wesley Publishing Company, ReadingUhlig, H., On singular Wishart and singular multivariate beta distributions (1994) Ann. Stat., 22 (1), pp. 395-405Zhang, Y., The exact distribution of the Mooore Penrose inverse of X with a density (1985) Multivariate Analysis VI, pp. 633-635. , Krishnaiah PR, (ed), Elsevier Science, AmsterdamBulletin of the Iranian Mathematical SocietyGeneralised inverseHausdorff measureJacobianLebesgue measureMatrix differentiationA Note About Measures, Jacobians and Moore–Penrose InverseArticleinfo:eu-repo/semantics/articlehttp://purl.org/coar/version/c_970fb48d4fbd8a85http://purl.org/coar/resource_type/c_6501http://purl.org/coar/resource_type/c_2df8fbb1Díaz-García, J.A., Facultad de Zootecnia y Ecología, Universidad Autónoma de Chihuahua, Periférico Francisco R. Almada Km 1, Zootecnia, Chihuahua, Chihuahua 33820, Mexico; Caro-Lopera, F.J., Faculty of Basic Sciences, Universidad de Medellín, Carrera 87 No. 30-65, of. 4-216, Medellín, Colombiahttp://purl.org/coar/access_right/c_16ecDíaz-García J.A.Caro-Lopera F.J.11407/5647oai:repository.udem.edu.co:11407/56472021-02-02 11:01:16.631Repositorio Institucional Universidad de Medellinrepositorio@udem.edu.co

A Note About Measures, Jacobians and Moore–Penrose Inverse

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