Hostname: page-component-77c89778f8-cnmwb Total loading time: 0 Render date: 2024-07-22T05:36:52.787Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Eigenvector belonging to the Maximal Root of a Non-negative Matrix

Published online by Cambridge University Press:  20 January 2009

A. M. Ostrowski
Rights & Permissions [Opens in a new window]

Extract

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.

By a theorem of Perron, a non-negative irreducible (n × n) matrix A = (aμν) has a positive fundamental root σ, the “ maximal root of A ”, such that the moduli of all other eigenvalues of A do not exceed σ. If we put

σ lies between R and r. Since σ is not changed if A is transformed by a positive diagonal matrix D(p1,…pn, σ lies also between the expressions

By a theorem of Frobenius, to σ as an eigenvalue of A belongs a positive eigenvector ξ, = (x1 …, xn), satisfying

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 12 , Issue 2 , December 1960 , pp. 107 - 112
Copyright
Copyright © Edinburgh Mathematical Society 1960

References

REFERENCES

(1)Brauer, A. T., The theorems of Ledermann and Ostrowski on positive matrices, Duke Math. J., 24 (1957), 265–274.CrossRefGoogle Scholar
(2)Ledermann, W., Bounds for the greatest latent root of a positive matrix, Journal London Math. Soc, 25 (1950), 265–268.CrossRefGoogle Scholar
(3)Ostrowski, A. M., Bounds for the greatest latent root of a positive matrix, Journal London Math. Soc, 27 (1952), 253–256.CrossRefGoogle Scholar
(4)Schneider, H., Note on the fundamental theorem on irreducible non-negative matrices, Proc. Edinburgh Math. Soc, 11 (2), 127–130.CrossRefGoogle Scholar