## A simple approach to the Perron-Frobenius theory for positive operators on general partially-ordered finite-dimensional linear spaces

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- by Werner C. Rheinboldt and James S. Vandergraft PDF
- Math. Comp.
**27**(1973), 139-145 Request permission

## Abstract:

This paper presents simple proofs of the principal results of the Perron-Frobenius theory for linear mappings on finite-dimensional spaces which are nonnegative relative to a general partial ordering on the space. The principal tool for these proofs is an application of the theory of norms in finite dimensions to the study of order inequalities of the form $Ax \leqq \alpha x,x \geqq 0$ where $A \geqq 0$. This approach also permits the derivation of various inclusion and comparison theorems.## References

- Garrett Birkhoff,
*Linear transformations with invariant cones*, Amer. Math. Monthly**74**(1967), 274–276. MR**214605**, DOI 10.2307/2316020 - Erich Bohl,
*Eigenwertaufgaben bei monotonen Operatoren und Fehlerabschätzungen für Operatorgleichungen*, Arch. Rational Mech. Anal.**22**(1966), 313–332 (German). MR**234305**, DOI 10.1007/BF00285424 - Ky Fan,
*Topological proofs for certain theorems on matrices with non-negative elements*, Monatsh. Math.**62**(1958), 219–237. MR**95856**, DOI 10.1007/BF01303967
G. Frobenius, "Über Matrizen aus positiven Elementen," - Alston S. Householder,
*The theory of matrices in numerical analysis*, Blaisdell Publishing Co. [Ginn and Co.], New York-Toronto-London, 1964. MR**0175290** - L. Kantorovitch,
*The method of successive approximations for functional equations*, Acta Math.**71**(1939), 63–97. MR**95**, DOI 10.1007/BF02547750 - M. G. Kreĭn and M. A. Rutman,
*Linear operators leaving invariant a cone in a Banach space*, Uspehi Matem. Nauk (N. S.)**3**(1948), no. 1(23), 3–95 (Russian). MR**0027128** - Oskar Perron,
*Zur Theorie der Matrices*, Math. Ann.**64**(1907), no. 2, 248–263 (German). MR**1511438**, DOI 10.1007/BF01449896 - Helmut H. Schaefer,
*Topological vector spaces*, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1966. MR**0193469** - H. Schneider and R. E. L. Turner,
*Positive eigenvectors of order-preserving maps*, J. Math. Anal. Appl.**37**(1972), 506–515. MR**288557**, DOI 10.1016/0022-247X(72)90292-2 - P. Stein and R. L. Rosenberg,
*On the solution of linear simultaneous equations by iteration*, J. London Math. Soc.**23**(1948), 111–118. MR**28682**, DOI 10.1112/jlms/s1-23.2.111 - P. Stein,
*Some general theorems on iterants*, J. Research Nat. Bur. Standards**48**(1952), 82–83. MR**0047001** - James S. Vandergraft,
*Spectral properties of matrices which have invariant cones*, SIAM J. Appl. Math.**16**(1968), 1208–1222. MR**244284**, DOI 10.1137/0116101 - James S. Vandergraft,
*Applications of partial orderings to the study of positive definiteness, monotonicity, and convergence of iterative methods for linear systems*, SIAM J. Numer. Anal.**9**(1972), 97–104. MR**309971**, DOI 10.1137/0709011 - Richard S. Varga,
*Matrix iterative analysis*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR**0158502**

*S.-B. Deutsch. Akad. Wiss. Berlin*, v. 1908, pp. 471-476. G. Frobenius, "Über Matrizen aus positiven Elementen. II,"

*S.-B. Deutsch. Akad. Wiss. Berlin*, v. 1909, pp. 514-518. G. Frobenius, "Über Matrizen aus nicht negativen Elementen,"

*S.-B. Deutsch. Akad. Wiss. Berlin*, v. 1912, pp. 456-477.

## Additional Information

- © Copyright 1973 American Mathematical Society
- Journal: Math. Comp.
**27**(1973), 139-145 - MSC: Primary 15A48
- DOI: https://doi.org/10.1090/S0025-5718-1973-0325650-4
- MathSciNet review: 0325650