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Journal of Applied Mathematics
Volume 2016 (2016), Article ID 4590817, 3 pages
http://dx.doi.org/10.1155/2016/4590817
Research Article

A Note on the vec Operator Applied to Unbalanced Block-Structured Matrices

Institute for Biodiversity and Ecosystem Dynamics, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, Netherlands

Received 25 July 2016; Accepted 3 October 2016

Academic Editor: Frédéric Guichard

Copyright © 2016 Hal Caswell and Silke F. van Daalen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. H. V. Henderson and S. R. Searle, “The vec-permutation matrix, the vec operator and Kronecker products: a review,” Linear and Multilinear Algebra, vol. 9, no. 4, pp. 271–288, 1981. View at Publisher · View at Google Scholar · View at MathSciNet
  2. R. H. Koning, H. Neudecker, and T. Wansbeek, “Block Kronecker products and the vecb operator,” Linear Algebra and Its Applications, vol. 149, pp. 165–184, 1991. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. I. Ojeda, “Kronecker square roots and the block vec matrix,” The American Mathematical Monthly, vol. 122, no. 1, pp. 60–64, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  4. J. R. Magnus and H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics, John Wiley & Sons, New York, NY, USA, 1988. View at MathSciNet
  5. H. Caswell, “Sensitivity and elasticity of density-dependent population models,” Journal of Difference Equations and Applications, vol. 15, no. 4, pp. 349–369, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. H. Caswell, “Perturbation analysis of continuous-time absorbing Markov chains,” Numerical Linear Algebra with Applications, vol. 18, no. 6, pp. 901–917, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. H. Caswell, “Sensitivity analysis of discrete Markov chains via matrix calculus,” Linear Algebra and Its Applications, vol. 438, no. 4, pp. 1727–1745, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. W. E. Roth, “On direct product matrices,” Bulletin of the American Mathematical Society, vol. 40, no. 6, pp. 461–468, 1934. View at Publisher · View at Google Scholar · View at MathSciNet
  9. M. Iosifescu, Finite Markov Processes and Their Applications, Wiley-Interscience, New York, NY, USA, 1980. View at MathSciNet
  10. H. Caswell, “Stage, age and individual stochasticity in demography,” Oikos, vol. 118, no. 12, pp. 1763–1782, 2009. View at Publisher · View at Google Scholar · View at Scopus
  11. F. R. Gantmacher, Matrix Theory, Volume 2, Chelsea, New York, NY, USA, 1959.