Hostname: page-component-7479d7b7d-8zxtt Total loading time: 0 Render date: 2024-07-12T04:21:01.721Z Has data issue: false hasContentIssue false
  • English
  • Français

Derivatives of the spectral radius as a function of non-negative matrix elements

Published online by Cambridge University Press:  24 October 2008

Joel E. Cohen
Joel E. Cohen
Affiliation:
The Rockefeller University, New York
Get access Rights & Permissions [Opens in a new window]

Extract

Let A = (aij) be a non-negative n × n matrix, that is, aij ≥ 0, i, j = 1, …, n; n > 1. We write A ≥ 0. Let r = r(A) be the spectral radius of A; assume r > 0 throughout to avoid trivial cases. Let be the mth derivative of r with respect to the element aij, all other elements of A being held constant.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 83 , Issue 2 , March 1978 , pp. 183 - 190
Copyright
Copyright © Cambridge Philosophical Society 1978

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Cohen, J. E.Ergodicity of age structure in populations with Markovian vital rates. III. Finite-state moments and growth rate; illustration. Advances in Appl. Probability 9 (1977).Google Scholar
(2)Faddeeva, V. N.Computational methods of linear algebra (Trans. Benster, Curtis D.. New York: Dover, 1959).Google Scholar
(3)Gantmacher, F. R.The theory of matrices, vol. 2 (New York: Chelsea Publishing Co., 1960).Google Scholar
(4)Kingman, J. F. C.A convexity property of positive matrices. Quart. J. Math. Oxford (2), 12 (1961), 283–284.CrossRefGoogle Scholar
(5)Lax, P. D.Differential equations, difference equations and matrix theory. Comm. Pure Appl. Math. 11 (1958), 175–194.CrossRefGoogle Scholar
(6)Leslie, P. H.On the use of matrices in certain population mathematics. Biometrika 33 (1945), 183–212.CrossRefGoogle ScholarPubMed
(7)Muir, T.A treatise on the theory of determinants (Rev. W. H. Metzler; privately published, Albany, New York, 1930, reprinted by Dover, New York).Google Scholar
(8)Seneta, E.Non-negative matrices: an introduction to theory and applications (London: George Allen and Unwin, 1973).Google Scholar
(9)Stouffer, E. B.On the independence of principal minors of determinants. Trans. Amer. Math. Soc. 26 (1924), 356–368.CrossRefGoogle Scholar
(10)Stouffer, E. B.Expression for a determinant in terms of five minors. Amer. Math. Monthly 39 (1932), 165–166.CrossRefGoogle Scholar