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Note on the Fundamental Theorem on Irreducible Non-Negative Matrices

Published online by Cambridge University Press:  20 January 2009

Hans Schneider
Hans Schneider
Affiliation:
The Queen's University, Belfast
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Let A = [aij] be an n-th order irreducible non-negative matrix. As is very well-known, the matrix A has a positive characteristic root ρ (provided that n>l), which is simple and maximal in the sense that every characteristic root λ satisfies |λ| ≤ρ, and the characteristic vector x belonging to ρ may be chosen positive. These results, originally due to Frobenius, have been proved by Wielandt (4) by means of a strikingly simple basic idea. Recently, a variant of Wielandt's proof has been given by Householder (2).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 11 , Issue 2 , November 1958 , pp. 127 - 130
Copyright
Copyright © Edinburgh Mathematical Society 1958

References

REFERENCES

(1) Bratuer, A., The theorems of Ledermann and Ostrowski on positive matrices, Duke Math. J., 24 (1957), 265–274.Google Scholar
(2) Householder, A. S., On the convergence of matrix iterations, Oak Ridge National Laboratory, Physics No. 1883 (1955).Google Scholar
(3) Ostrowski, A., Bounds for the greatest latent root of a positive matrix, J. London Math. Soc., 27 (1952), 253–256.CrossRefGoogle Scholar
(4) Wiblandt, H., Unzerlegbare, nicht negative Matrizen, Math. Zeitschrift, 52 (1950), 642–8.CrossRefGoogle Scholar