The Perron generalized eigenspace and the spectral cone of a cone-preserving map

Linear Algebra and Its Applications

Volumn 393, Issue 1-3, 2004, Pages 375-429

  Tam, Bit Shun ^a  

^a TAMKANG UNIVERSITY   (Taiwan)

Author keywords

Cone preserving map; Height characteristic; Level characteristics; Majorization relation; Nonnegative matrix; Perron generalized eigenspace; Preferred basis theorem; Principal component; Semi distinguished invariant face; Spectral cone

Indexed keywords

EIGENVALUES AND EIGENFUNCTIONS; MATHEMATICAL MODELS; MATRIX ALGEBRA; SET THEORY; THEOREM PROVING; VECTORS;

CONE-PRESERVING MAP; HEIGHT CHARACTERISTICS; LEVEL CHARACTERISTICS; MAJORIZATION RELATIONS; NONNEGATIVE MATRIX; PRINCIPAL COMPONENTS;

COMBINATORIAL MATHEMATICS;

EID: 7444267410     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.laa.2004.08.020     Document Type: Article

References (38)

1. G.P. Barker, On matrices having an invariant cone, Czech. Math. J. 22 (1972) 49-68.
2. G. Birkhoff, Linear transformations with invariant cones, Amer. Math. Monthy 74 (1967) 274-276.
3. R. Bru, M. Neumann, Nonnegative Jordan basis, Linear and Multilinear Algebra 23 (1988) 95-109.
4. A. Berman, M. Neumann, R.J. Stern, Nonnegative Matrices in Dynamic Systems, Wiley, New York, 1989
5. R. Bru, L. Rodman, H. Schneider, Extensions of Jordan bases for invariant subspaces of a matrix, Linear Algebra Appl. 150 (1991) 209-225.
6. L. Elsner, Quasimonotonie und Ungliechungen in halbgeordneten Raumen, Linear Algebra Appl. 8 (1974) 249-261.
7. S. Friedland, H. Schneider, The growth of powers of a nonnegative matrix, SIAM J. Algebraic Discrete Methods 1 (1980) 185-200.
8. P. Gritzmann, V. Klee, B.S. Tam, Cross-positive matrices revisited, Linear Algebra Appl. 223/224 (1995) 285-305.
9. R.E. Hartwig, M. Neumann, N.J. Rose, An algebraic-analytic approach to nonnegative basis, Linear Algebra Appl. 133 (1990) 77-88.
10. D. Hershkowitz, Peak characteristic and nonnegative Signature, Linear Algebra Appl. 147 (1991) 55-73.
11. D. Hershkowitz, Paths in directed graphs and spectral properties of matrices, Linear Algebra Appl. 212/213 (1994) 309-337.
12. D. Hershkowitz, The combinatorial structure of generalized eigenspaces - from nonnegative matrices to general matrices, Linear Algebra Appl. 302/303 (1999) 173-191.
13. D. Hershkowitz, H. Schneider, On the generalized nullspace of M-matrices and Z-matrices, Linear Algebra Appl. 106 (1988) 5-23.
14. D. Hershkowitz, H. Schneider, Height bases, level bases, and the equality of the height and the level characteristic of an M-matrix, Linear and Multilinear Algebra 25 (1989) 149-171.
15. D. Hershkowitz, H. Schneider, Combinatorial bases, derived Jordan sets, and the equality of the height and the level characteristics of an M-matrix, Linear and Multilinear Algebra 29 (1991) 21-42.
16. D. Hershkowitz, H. Schneider, On the existence of matrices with prescribed height and level characteristics, Israel J. Math. 75 (1991) 105-117.
17. C.R. Johnson, E.A. Schreiner, The relationship between AB and BA, Amer. Math. Monthly 103 (1996) 578-582.
18. P. Lancaster, M. Tismenetsky, The Theory of Matrices, second ed., Academic Press, New York, 1985.
19. J.J. McDonald, M. Neumann, H. Schneider, Resolvent of mains M-matrices and splittings of M-matrices, Linear Algebra Appl. 195 (1993) 17-33.
20. A.W. Marshall, I. Olkin, Inequalities Theory of Majorization and its Applications, Academic Press, 1979.
21. C.D. Meyer Jr, M.W. Stadelmaier, Singular M-matrices and inverse positivity, Linear Algebra Appl. 22 (1978) 139-156.
22. M. Neumann, H. Schneider, Principal components of minus M-matrices, Linear and Multilinear Algebra 32 (1992) 131-148.
23. M. Neumann, H. Schneider, Corrections and additions to: principal components of minus M-matrices, Linear and Multilinear Algebra 36 (1993) 147-149.
24. M. Neumann, H. Schneider, Algorithms for computing bases for the Perron eigenspace with prescribed nonnegativity and combinatorial properties, SIAM J. Matrix Anal. Appl. 15 (1994) 578-591.
25. D.J. Richman, H. Schneider, On the singular graph and the Weyr characteristic of an M-matrix, Aequationes Math. 17 (1978) 208-234.
26. R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.
27. U.G. Rothblum, Algebraic eigenspaces of non-negative matrices, Linear Algebra Appl. 12 (1975) 281-292.
28. H. Schneider, M. Vidyasagar, Cross-positive matrices, SIAM J. Numer. Anal. 7 (1970) 508-519.
29. H. Schneider, Geometric conditions for the existence of positive eigenvalues of matrices, Linear Algebra Appl. 38 (1981) 253-271.
30. H. Schneider, The influence of the marked reduced graph of a nonnegative matrix on the Jordan form and on related properties: a survey, Linear Algebra Appl. 84 (1986) 161-189.
31. B.S. Tam, On the duality operator of a convex cone, Linear Algebra Appl. 64 (1985) 33-56.
32. B.S. Tam, On the distinguished eigenvalues of a cone-preserving map, Linear Algebra Appl. 131 (1990) 17-37.
33. B.S. Tam, On the structure of the cone of positive operators, Linear Algebra Appl. 167 (1992) 65-86.
34. B.S. Tam, A cone-theoretic approach to the spectral theory of positive linear operators: the finite-dimensional case, Taiwanese J. Math. 5 (2001) 207-277.
35. B.S. Tam, H. Schneider, On the core of a cone-preserving map, Trans. Amer. Math. Soc. 343 (1994) 479-524.
36. B.S. Tam, H. Schneider, On the invariant faces associated with a cone-preserving map, Trans. Amer. Math. Soc. 353 (2001) 209-245.
37. B.S. Tam, H. Schneider, Linear equations over cones and Collatz-Wielandt numbers, Linear Algebra Appl. 363 (2003) 295-332.
38. B.S. Tam, S.F. Wu, On the Collatz-Wielandt sets associated with a cone-preserving map, Linear Algebra Appl. 125 (1989) 77-95.