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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
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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

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