Geodesic
Falseness of the finiteness property of the spectral subradius
International Journal of Applied Mathematics and Computer Science, Tome 17 (2007) no. 2, pp. 173-178.

Voir la notice de l'article provenant de la source Library of Science

We prove that there exist infinitely may values of the real parameter α for which the exact value of the spectral subradius of the set of two matrices (one matrix with ones above and on the diagonal and zeros elsewhere, and one matrix with α below and on the diagonal and zeros elsewhere, both matrices having two rows and two columns) cannot be calculated in a finite number of steps. Our proof uses only elementary facts from the theory of formal languages and from linear algebra, but it is not constructive because we do not show any explicit value of α that has described property. The problem of finding such values is still open.
Keywords: finiteness property, spectral subradius
Mots-clés : właściwość skończoności, podpromień spektralny 
@article{IJAMCS_2007_17_2_a3,
     author = {Czornik, A. and Jurga\'s, P.},
     title = {Falseness of the finiteness property of the spectral subradius},
     journal = {International Journal of Applied Mathematics and Computer Science},
     pages = {173--178},
     publisher = {mathdoc},
     volume = {17},
     number = {2},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IJAMCS_2007_17_2_a3/}
}
TY  - JOUR
AU  - Czornik, A.
AU  - Jurgaś, P.
TI  - Falseness of the finiteness property of the spectral subradius
JO  - International Journal of Applied Mathematics and Computer Science
PY  - 2007
SP  - 173
EP  - 178
VL  - 17
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IJAMCS_2007_17_2_a3/
LA  - en
ID  - IJAMCS_2007_17_2_a3
ER  - 
%0 Journal Article
%A Czornik, A.
%A Jurgaś, P.
%T Falseness of the finiteness property of the spectral subradius
%J International Journal of Applied Mathematics and Computer Science
%D 2007
%P 173-178
%V 17
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IJAMCS_2007_17_2_a3/
%G en
%F IJAMCS_2007_17_2_a3
Czornik, A.; Jurgaś, P. Falseness of the finiteness property of the spectral subradius. International Journal of Applied Mathematics and Computer Science, Tome 17 (2007) no. 2, pp. 173-178. http://geodesic.mathdoc.fr/item/IJAMCS_2007_17_2_a3/

[1] Elsner L. (1995): The generalized spectral radius theorem: An analytic-geometric proof. - Linear Algebra Appl., Vol. 220, pp. 151-159.

[2] Blondel V.D., Theys J. and Vladimirov A.A. (2003): An elementary counterexample to the finitness conjecture. - SIAM J. Matrix Anal. Appl., Vol. 24, No. 4, pp. 963-970.

[3] Czornik A. (2005): On the generalized spectral subradius. - Linear Algebra Appl., Vol. 407, pp. 242-248.

[4] Horn R.A. and Johnson C.R. (1991): Topics in Matrix Analysis. - Cambridge, UK: Cambridge University Press.

[5] Horn R.A. and Johnson C.R. (1985): Matrix Analysis.- Cambridge, UK: Cambridge University Press.

[6] Lagarias J.C. and Wang Y. (1995): The finiteness conjecture for the generalized spectral radius of a set of matrices. - Linear Algebra Appl., Vol. 214, pp. 17-42.

[7] Tsitsiklis J. and Blondel V. (1997): The Lyapunov exponent and joint spectral radius of pairs of matrices are hard - when not impossible to compute and to approximate. - Math. Contr. Signals Syst., Vol. 10, pp. 31-40.