Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-02-03T22:52:19.482Z Has data issue: false hasContentIssue false
  • English
  • Français

On Bounded Matrices with Non-Negative Elements

Published online by Cambridge University Press:  20 November 2018

C. R. Putnam
C. R. Putnam*
Affiliation:
Purdue University
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.

It is known (Perron (10); Frobenius (5, 6)) that if A = (aik) is a finite matrix with elements aik ⩾ 0, then A has a real, nonnegative eigenvalue μ, satisfying μ =max|λ| where λ is in the spectrum of A, with a corresponding eigenvector x = (x1, … , xn) for which xi≥ 0. Moreover if aik > 0, then μ is a simple point of the spectrum with an eigenvector x (unique, except for constant multiples) with components xi ≥0. Much has been written on this and related issues; cf., for example, the recent papers (4, 12) wherein are given several references.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 10 , 1958 , pp. 587 - 591
Copyright
Copyright © Canadian Mathematical Society 1958

References

1. Bonsall, F. F. , Endomorphisms of partially ordered vector spaces, J. Lond. Math. Soc, 30 (1955), 133-144.Google Scholar
2. Bonsall, F. F., Endomorphisms of a partially ordered vector space without order unit, J. Lond. Math. Soc, 30 (1955), 144-153.Google Scholar
3. Bonsall, F. F., Linear operators in complete positive cones, Proc. Lond. Math. Soc, Ser. 3, 8(1958), 53-75.Google Scholar
4. Brauer, A., A new proof of theorems of Perron and Frobenius on non-negative matrices, I. Positive matrices, Duke Math. J., 24 (1957), 367–378.Google Scholar
5. Frobenius, G., Ueber Matrizen aus positiven Elementen, Sitzungsberichte Preuss. Akad. Wiss. (Berlin, 1908), 471-476; (1909), 514-518.Google Scholar
6. Frobenius, G., Ueber Matrizen aus nicht negakven Elementen, Sitzungsberichte Preuss. Akad. Wiss. (Berlin, 1912), 456-477.Google Scholar
7. Hartman, P. and Wintner, W., The spectra of Toeplitz's matrices, Amer. J. Math., 76 (1954), 867-882.Google Scholar
8. Krein, M. G. and Rutman, M. A., Linear operators leaving invariant a cone in a Banach space, Uspehi Matem, Nauk (N.S.) 3, no. 1 (23) (1948), 3-95; Amer. Math. Soc. Translation No. 26 (page references in paper refer to this translation).Google Scholar
9. Landau, E., Darstellung und Begrundung etniger neuerer Ergebnisse der Funkkonentheorie (Berlin, 1929).Google Scholar
10. Perron, O., Zur Theorie der Matrices, Math. Ann., 64 (1907), 248-263.Google Scholar
11. Rutman, M. A., Sur les opérateurs totalement continus linéaires laissant invariant un certain cone, Math. Sbornik (N.S.), 8, no. 1 (50) (1940), 77-96.Google Scholar
12. Samelson, H., On the Perron-Frobenius theorem, Michigan Math. J., 4 (1957), 57 59.Google Scholar