Results on measures of irreducibility and full indecomposability

Author:
D. J. Hartfiel

Journal:
Trans. Amer. Math. Soc. **202** (1975), 357-368

MSC:
Primary 15A48

DOI:
https://doi.org/10.1090/S0002-9947-1975-0364303-1

MathSciNet review:
0364303

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper develops a notion of th measure of irreducibility and th measure of full indecomposability. The combinatorial properties of these notions, as well as relationships between these notions, are explored. The results are then used in converting results on positive matrices into results on nonnegative matrices.

**[1]**Richard Bellman,*On a quasi-linear equation*, Canad. J. Math.**8**(1956), 198-202. MR**17**, 1093. MR**0077782 (17:1093e)****[2]**R. A. Brualdi and Hazel Perfect,*Extension of partial diagonals of matrices*. I, Monatsh. Math.**75**(1971), 385-397. MR**0309966 (46:9069)****[3]**R. A. Brualdi, S. V. Parter and H. Schneider,*The diagonal equivalence of a non-negative matrix to a stochastic matrix*, J. Math. Anal. Appl.**16**(1966), 31-50. MR**34**#5844. MR**0206019 (34:5844)****[4]**D. J. Hartfiel,*A simplified form for nearly reducible and nearly decomposable matrices*, Proc. Amer. Math. Soc.**24**(1970), 388-393. MR**40**#5636; erratum,**41**, p. 1965. MR**0252415 (40:5635)****[5]**-,*Bounds for eigenvalues and eigenvectors of a non-negative matrix which involve a measure of irreducibility*, SIAM J. Appl. Math.**24**(1973), 6-8. MR**0325648 (48:3995)****[6]**-,*Concerning infinite products of matrices*, SIAM J. Appl. Math.**26**(1974), 297-301. MR**0352144 (50:4631)****[7]**M. Lewin,*On nonnegative matrices*, Pacific J. Math.**36**(1971), 753-759. MR**44**#2772. MR**0285554 (44:2772)****[8]**M. S. Lynn and W. P. Timlake,*Bounds for Perron eigenvectors and subdominant eigenvalues of positive matrices*, Linear Algebra and Appl.**2**(1969), 143-152. MR**39**#1477. MR**0240123 (39:1477)****[9]**Marvin Marcus and Henryk Minc,*A survey of matrix theory and matrix inequalities*, Allyn and Bacon, Inc., Boston, Mass., 1964. MR**29**#112. MR**0162808 (29:112)****[10]**Henryk Minc,*On the maximal eigenvector of a positive matrix*, SIAM J. Numer. Anal.**7**(1970), 424-427. MR**42**#1320. MR**0266414 (42:1320)****[11]**A. M. Ostrowski,*On the eigenvector belonging to the maximal root of a nonnegative matrix*, Rep. 145, Mathematics Research Center, U. S. Army, Madison, Wisconsin, 1960.**[12]**Richard Sinkhorn and Paul Knopp,*Problems involving diagonal products in non-negative matrices*, Trans. Amer. Math. Soc.**136**(1969), 67-75. MR**38**#2151. MR**0233830 (38:2151)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
15A48

Retrieve articles in all journals with MSC: 15A48

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1975-0364303-1

Keywords:
Fully indecomposable,
irreducible,
bounds on eigenvalues and eigenvectors,
Markov chains

Article copyright:
© Copyright 1975
American Mathematical Society