Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-23T21:23:26.108Z Has data issue: false hasContentIssue false
  • English
  • Français

A CANCELLATION PROPERTY OF THE MOORE–PENROSE INVERSE OF TRIPLE PRODUCTS

Part of: Basic linear algebra

Published online by Cambridge University Press:  01 February 2009

TOBIAS DAMM  and
HARALD K. WIMMER
TOBIAS DAMM*
Affiliation:
Fachbereich Mathematik, TU Kaiserslautern, D-67663 Kaiserslautern, Germany (email: [email protected])
HARALD K. WIMMER
Affiliation:
Mathematisches Institut, Universität Würzburg, D-97074 Würzburg, Germany (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the matrix equation C(BXC)†B=X†, where X† denotes the Moore–Penrose inverse. We derive conditions for the consistency of the equation and express all its solutions using singular vectors of B and C. Applications to compliance matrices in molecular dynamics, to mixed reverse-order laws of generalized inverses and to weighted Moore–Penrose inverses are given.

Keywords

Moore–Penrose generalized inversecompliance matrixreverse-order propertyweighted generalized inverseWedderburn–Guttman theoremmatrix equations

MSC classification

Secondary: 15A09: Matrix inversion, generalized inverses
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 86 , Issue 1 , February 2009 , pp. 33 - 44
Copyright
Copyright © Australian Mathematical Society 2009

References

[1]Ben-Israel, A. and Greville, Th. N. E., Generalized Inverses, 2nd edn (Springer, Berlin, 2003).Google Scholar
[2]Brandhorst, K., ‘Computations of non-covalent interactions in quantum mechanics’, Thesis, Department of Life Sciences, Technical University of Braunschweig, 2006 (in German).Google Scholar
[3]Campbell, St. L. and Meyer, C. D. Jr, Generalized Inverses of Linear Transformations (Dover Publications, New York, 1991).Google Scholar
[4]Grunenberg, J. and Goldberg, N., ‘How strong is the Gallium≡Gallium triple bond? Theoretical compliance matrices as a probe for intrinsic bond strengths’, J. Am. Chem. Soc. 122 (2000), 6045–6047.CrossRefGoogle Scholar
[5]Grunenberg, J., Streubel, R., von Frantzius, G. and Marten, W., ‘The strongest bond in the universe? Accurate calculation of compliance matrices for the ions N2H+, HCO+, and HOC+’, J. Chem. Phys. 119 (2003), 165–169.CrossRefGoogle Scholar
[6]Martínez-Torres, E., López-Gonzáles, J. J. and Fernández-Gómez, M., ‘Unambiguous formalism of molecular vibrations: use of redundant coordinates and canonical matrices’, J. Chem. Phys. 110 (1999), 3302–3308.CrossRefGoogle Scholar
[7]Mazko, A. G., ‘Semiinversion and properties of matrix invariants’, Ukrain. Mat. Zh. 40 (1988), 525–528 (Russian).Google Scholar
[8]Peng, C., Ayala, P. Y., Schlegel, H. B. and Frisch, M. J., ‘Using redundant internal coordinates to optimize equilibrium geometries and transition states’, J. Comp. Chem. 17(1) (1995), 49–56.3.0.CO;2-0>CrossRefGoogle Scholar
[9]Takane, Y., Tian, Y. and Yanai, H., ‘On constrained generalized inverses of matrices and their properties’, Ann. Inst. Stat. Math. 59 (2007), 807–820.CrossRefGoogle Scholar
[10]Takane, Y. and Yanai, H., ‘On the Wedderburn–Guttman theorem’, Linear Algebra Appl. 410 (2005), 267–278.CrossRefGoogle Scholar
[11]Takane, Y. and Yanai, H., ‘Alternative characterizations of the extended Wedderburn–Guttman theorem’, Linear Algebra Appl. 422 (2007), 701–711.CrossRefGoogle Scholar
[12]Tian, Y., ‘The reverse-order law (AB)†=B †(A †ABB †)†A † and its equivalent equalities’, J. Math. Kyoto Univ. 45 (2005), 841–850.Google Scholar
[13]Tian, Y., ‘The equivalence between (AB)†=B †A † and other mixed-type reverse-order laws’, Internat. J. Math. Ed. Sci. Tech. 37 (2006), 331–339.CrossRefGoogle Scholar
[14]Tian, Y. and Liu, Y., ‘On a group of mixed-type reverse-order laws for generalized inverses of a triple matrix product with applications’, Electron. J. Linear Algebra 16 (2007), 73–89.CrossRefGoogle Scholar
[15]Watson, J. K. G., ‘A comment on the use of redundant vibrational constants’, J. Mol. Struct. 695–696 (2004), 71–75.CrossRefGoogle Scholar