Upper bound for the non-maximal eigenvalues of irreducible nonnegative matrices

Xiao-Dong Zhang; Rong Luo

Czechoslovak Mathematical Journal (2002)

  • Volume: 52, Issue: 3, page 537-544
  • ISSN: 0011-4642

Abstract

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We present a lower and an upper bound for the second smallest eigenvalue of Laplacian matrices in terms of the averaged minimal cut of weighted graphs. This is used to obtain an upper bound for the real parts of the non-maximal eigenvalues of irreducible nonnegative matrices. The result can be applied to Markov chains.

How to cite

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Zhang, Xiao-Dong, and Luo, Rong. "Upper bound for the non-maximal eigenvalues of irreducible nonnegative matrices." Czechoslovak Mathematical Journal 52.3 (2002): 537-544. <http://eudml.org/doc/30722>.

@article{Zhang2002,
abstract = {We present a lower and an upper bound for the second smallest eigenvalue of Laplacian matrices in terms of the averaged minimal cut of weighted graphs. This is used to obtain an upper bound for the real parts of the non-maximal eigenvalues of irreducible nonnegative matrices. The result can be applied to Markov chains.},
author = {Zhang, Xiao-Dong, Luo, Rong},
journal = {Czechoslovak Mathematical Journal},
keywords = {eigenvalue; irreducible nonnegative matrix; averaged minimal cut; eigenvalue; irreducible nonnegative matrix; averaged minimal cut; Laplacian matrices; weighted graphs; Markov chains},
language = {eng},
number = {3},
pages = {537-544},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Upper bound for the non-maximal eigenvalues of irreducible nonnegative matrices},
url = {http://eudml.org/doc/30722},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Zhang, Xiao-Dong
AU - Luo, Rong
TI - Upper bound for the non-maximal eigenvalues of irreducible nonnegative matrices
JO - Czechoslovak Mathematical Journal
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 52
IS - 3
SP - 537
EP - 544
AB - We present a lower and an upper bound for the second smallest eigenvalue of Laplacian matrices in terms of the averaged minimal cut of weighted graphs. This is used to obtain an upper bound for the real parts of the non-maximal eigenvalues of irreducible nonnegative matrices. The result can be applied to Markov chains.
LA - eng
KW - eigenvalue; irreducible nonnegative matrix; averaged minimal cut; eigenvalue; irreducible nonnegative matrix; averaged minimal cut; Laplacian matrices; weighted graphs; Markov chains
UR - http://eudml.org/doc/30722
ER -

References

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  1. Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York-San Francisco-London, 1979; SIAM, Philadelphia, 1994. (1979; SIAM, Philadelphia, 1994) MR1298430
  2. An estimate for the non-stochastic eigenvalues of doubly stochastic matrices, Linear Algebra Appl. 214 (1995), 133–143. (1995) MR1311633
  3. On the structure of stochastic matrices with a subordinant eigenvalue near  1, Linear Algebra Appl. 272 (1998), 193–203. (1998) MR1489387
  4. Inequalities: Theory of Majorization and Applications, Academic Press, New York, 1979. (1979) MR0552278
  5. Laplacian matrices of graphs: a survey, Linear Algebra Appl. 197–198 (1994), 143–176. (1994) Zbl0802.05053MR1275613
  6. A survey of graph laplacians, Linear and Multilinear Algebra 39 (1995), 19–31. (1995) Zbl0832.05081MR1374468
  7. 10.1016/0012-365X(92)90288-Q, Discrete Math. 109 (1992), 171–183. (1992) Zbl0783.05073MR1192380DOI10.1016/0012-365X(92)90288-Q
  8. Note on the fundamental theorem on irreducible nonnegative matrices, Proc. Edinburgh Math. Soc. 12 (1960/1961), 107–112. (1960/1961) 

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