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Research Article

On the Moore–Penrose inverse of a sum of matrices

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Pages 133-149 | Received 29 Aug 2021, Accepted 14 Dec 2021, Published online: 06 Jan 2022

References

  • Penrose R. A generalized inverse for matrices. Proc Cambridge Philos Soc. 1955;51:406–413.
  • Bernstein DS. Scalar, vector, and matrix mathematics: theory, facts, and formulas. Princeton (NJ): Princeton University Press; 2018.
  • Hung C-H, Markham TL. The Moore–Penrose inverse of a sum of matrices. J Austral Math Soc Ser A. 1977;24:385–392.
  • Cline RE. Representation of the generalized inverse of sums of matrices. SIAM J Numer Anal Ser B. 1965;2:99–114.
  • Drazin MP. Natural structures on semigroups with involution. Bull Amer Math Soc. 1978;84:139–141.
  • Hartwig RE, Styan GPH. On some characterizations of the ‘star’ partial ordering for matrices and rank subtractivity. Linear Algebra Appl. 1986;82:145–161.
  • Hartwig RE. How to partially order regular elements. Math Japon. 1980;25:1–13.
  • Nambooripad KSS. The natural partial order on a regular semigroup. Proc Edinburgh Math Soc. 1980;23:249–260.
  • Marsaglia G, Styan GPH. Equalities and inequalities for ranks of matrices. Linear Multilinear Algebra. 1974;2:269–292.
  • Cline RE, Funderlic RE. The rank of a difference of two matrices and associated generalized inverses. Linear Algebra Appl. 1979;24:185–215.
  • Mitra SK. A pair of simultaneous linear matrix equations A1XB1=C1, A2XB2=C2 and a matrix programming problem. Linear Algebra Appl. 1990;131:107–123.
  • Baksalary OM, Trenkler G. Column space equalities for orthogonal projectors. Appl Math Comput. 2009;212:519–529.
  • Baksalary OM, Trenkler G. Functions of orthogonal projectors involving the Moore–Penrose inverse. Comput Math Appl. 2010;59:764–778.
  • Ben-Israel A, Greville TNE. Generalized inverses: theory and applications. 2nd ed. New York (NY): Springer; 2003.
  • Beylkin G, Mohlenkamp MJ, Pérez F. Approximating a wavefunction as an unconstrained sum of Slater determinants. J Math Phys. 2008;49:032107.
  • Saha R, Aluri PK. A perturbative analysis of synchrotron spectral index variation over the microwave sky. Astrophys J. 2016;829:113.
  • Saha R, Prunet S, Jain P, et al. CMB anisotropy power spectrum using linear combinations of WMAP maps. Phys Rev D. 2008;78:023003.
  • Liu LP, Jiang Y, Zhou ZH. Least square incremental linear discriminant analysis. In: Wang W, Kargupta H, Ranka S, et al., editors. Proceedings of the 9th IEEE International Conference on Data Mining; 2009 Dec 6–9; Miami Beach, FL, USA. Washington, DC: IEEE Computer Society; 2009. p. 298–306.
  • Wang Q, Zhang L. Least squares online linear discriminant analysis. Expert Syst Appl. 2012;39:1510–1517.
  • Hu S, Dezhong Y, Bringas-Vega ML, et al. The statistics of EEG unipolar references: derivations and properties. Brain Topogr. 2019;32:696–703.
  • Hu S, Dezhong Y, Valdes-Sosa PA. Unified Bayesian estimator of EEG reference at infinity: rREST (regularized reference electrode standardization technique). Front Neurosci. 2018;12:297.
  • Steerneman T, van Perlo-ten Kleij F. Properties of the matrix A−XY∗. Linear Algebra Appl. 2005;410:70–86.
  • Trenkler G. On a generalisation of the covariance matrix of the multinomial distribution. In: Heijmans RDH, Pollock DSG, Satorra A, editors. Innovations in Multivariate Statistical Analyses: A Festschrift for Heinz Neudecker. Dordrecht: Springer; 2000. p. 67–73.
  • Baksalary JK, Baksalary OM, Trenkler G. A revisitation of formulae for the Moore–Penrose inverse of modified matrices. Linear Algebra Appl. 2003;372:207–224.
  • Baksalary OM, Trenkler G. On formulae for the Moore–Penrose inverse of a columnwise partitioned matrix. Appl Math Comput. 2021;403:125913.
  • Groß J. On oblique projection, rank additivity and the Moore–Penrose inverse of the sum of two matrices. Linear Multilinear Algebra. 1999;46:265–275.
  • Hartwig RE, Spindelböck K. Matrices for which A∗ and A† commute. Linear Multilinear Algebra. 1984;14:241–256.
  • Baksalary OM, Trenkler G. On disjoint range matrices. Linear Algebra Appl. 2011;435:1222–1240.
  • Trenkler G. On oblique and orthogonal projectors. In: Brown P, Liu S, Sharma D, editors. Proceedings of the International Statistics Workshop – Contributions to Probability and Statistics: Applications and Challenges; Singapore: World Scientific; 2006. p. 178–191.
  • Campbell SL, Meyer CD. Generalized inverses of linear transformations. SIAM: Philadelphia (PA); 2009.
  • Anderson Jr. WN, Kleindorfer GB, Kleindorfer PR, et al. Consistent estimates of the parameters of a linear system. Ann Math Stat. 1969;40:2064–2075.
  • Zhong D, Welsch W. The parallel sum of matrices and its application in geodetic adjustments. J Geod. 1997;71:171–175.
  • Baksalary OM, Styan GPH, Trenkler G. On a matrix decomposition of Hartwig and Spindelböck. Linear Algebra Appl. 2009;430:2798–2812.

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