# Falseness of the finiteness property of the spectral subradius

International Journal of Applied Mathematics and Computer Science (2007)

- Volume: 17, Issue: 2, page 173-178
- ISSN: 1641-876X

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topCzornik, Adam, and Jurgas, Piotr. "Falseness of the finiteness property of the spectral subradius." International Journal of Applied Mathematics and Computer Science 17.2 (2007): 173-178. <http://eudml.org/doc/207829>.

@article{Czornik2007,

abstract = {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.},

author = {Czornik, Adam, Jurgas, Piotr},

journal = {International Journal of Applied Mathematics and Computer Science},

keywords = {finiteness property; spectral subradius},

language = {eng},

number = {2},

pages = {173-178},

title = {Falseness of the finiteness property of the spectral subradius},

url = {http://eudml.org/doc/207829},

volume = {17},

year = {2007},

}

TY - JOUR

AU - Czornik, Adam

AU - Jurgas, Piotr

TI - Falseness of the finiteness property of the spectral subradius

JO - International Journal of Applied Mathematics and Computer Science

PY - 2007

VL - 17

IS - 2

SP - 173

EP - 178

AB - 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.

LA - eng

KW - finiteness property; spectral subradius

UR - http://eudml.org/doc/207829

ER -

## References

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- Czornik A. (2005): On the generalized spectral subradius. -Linear Algebra Appl., Vol.407, pp.242-248. Zbl1080.15008
- Horn R.A. and Johnson C.R. (1991): Topics in Matrix Analysis. - Cambridge, UK: Cambridge University Press. Zbl0729.15001
- Horn R.A. and Johnson C.R. (1985): Matrix Analysis. -Cambridge, UK: Cambridge University Press. Zbl0576.15001
- 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. Zbl0818.15007
- 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 Zbl0888.65044

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