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, 23:2 | Cite as

Inverse of sum of Kronecker products as a sum of Kronecker products

  • Abdelhalim NiatiEmail author
Review Article
  • 172 Downloads

Abstract

In the context of processing global navigation satellite system (GNSS) data by least squares adjustment, one may encounter a mathematical problem when inverting a sum of two Kronecker products. As a solution of this problem, we propose to invert this sum in the form of another sum of two Kronecker products. We present and demonstrate two mathematical formulas that enable us to achieve this task. We conclude from the demonstration that there is one condition for each formula to be checked before applying this proposed matrix inversion technique. In fact, these conditions restrict greatly the application of the formulas from being more general to the inversion problems of this kind. However, when applicable, the formulas obviously save computations in general and are very useful for large and fully populated matrices. In addition, this proposed matrix inversion technique shows several benefits when used in the processing of a single baseline with multi-frequency GNSS signals. These benefits are summarized in the following. First, the fully populated variance–covariance matrix of observations is easily inverted. Second, the computation of the normal matrix becomes easier as well since the blocks in both the design and weight matrix are all written in the form of Kronecker products. Third, this proposed matrix inversion technique contributes greatly to the computation of the variance–covariance matrix of estimates.

Keywords

Sum of Kronecker products Matrix inverse Processing GNSS data 

References

  1. Gene HG, Charles FVL (2013) Matrix computations, 4th edn. Johns Hopkins University Press, Baltimore, pp 681–746Google Scholar
  2. Henderson HV, Searle SR (1981) On deriving the inverse of a sum of matrices. SIAM Rev 23(1):53–60CrossRefGoogle Scholar
  3. Higham NJ (2002) Accuracy and stability of numerical algorithms, Chap. 14, 2nd edn. Society of Industrial and Applied Mathematics, Philadelphia (ISBN 0-89871-521-0) CrossRefGoogle Scholar
  4. Kedar S, Hajj GA, Wilson BD, Heflin MB (2003) The effect of the second order GPS ionospheric correction on receiver positions. Geophys Res Lett 30(16):1829.  https://doi.org/10.1029/2003GL017639 CrossRefGoogle Scholar
  5. Kim D, Langley RB (2001) Estimation of the stochastic model for long-baseline kinematic GPS applications. In: Proc ION NTM 2001, Institute of Navigation, Long Beach, 22–24 January, pp 586–595Google Scholar
  6. Laub AJ (2005) Matrix analysis for scientists and engineers. Society of Industrial and Applied Mathematics, Philadelphia (ISBN 0-89871-576-8)CrossRefGoogle Scholar
  7. Liu X (2002) A comparison of stochastic models for GPS single differential kinematic positioning. In: Proc ION GPS 2002, Institute of Navigation, Portland, 24–27 September, pp 1830–1841Google Scholar
  8. Miller KS (1981) On the inverse of the sum of matrices. Math Mag 54(2):67–72.  https://doi.org/10.2307/2690437 (Published by Mathematical Association of America) CrossRefGoogle Scholar
  9. Nardo A, Li B, Teunissen PJG (2015) Partial ambiguity resolution for ground and space-based applications in a GPS + Galileo scenario: a simulation study. Adv Space Res 57(1):30–45.  https://doi.org/10.1016/j.asr.2015.09.002 CrossRefGoogle Scholar
  10. Teunissen PJG, Jonkman NF, Tiberius CCJM (1998) Weighting GPS dual frequency observations: bearing the cross of cross-correlation. GPS Solut 2(2):28–37CrossRefGoogle Scholar
  11. Zhang H, Ding F (2013) On the Kronecker products and their applications. J Appl Math.  https://doi.org/10.1155/2013/296185. (Article ID 296185, Hindawi Publishing Corporation) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of GeodesyCenter of Space Techniques (CTS)OranAlgeria

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