Some properties of the spectral radius of a set of matrices

Adam Czornik; Piotr Jurgas

International Journal of Applied Mathematics and Computer Science (2006)

  • Volume: 16, Issue: 2, page 183-188
  • ISSN: 1641-876X

Abstract

top
In this paper we show new formulas for the spectral radius and the spectral subradius of a set of matrices. The advantage of our results is that we express the spectral radius of any set of matrices by the spectral radius of a set of symmetric positive definite matrices. In particular, in one of our formulas the spectral radius is expressed by singular eigenvalues of matrices, whereas in the existing results it is expressed by eigenvalues.

How to cite

top

Czornik, Adam, and Jurgas, Piotr. "Some properties of the spectral radius of a set of matrices." International Journal of Applied Mathematics and Computer Science 16.2 (2006): 183-188. <http://eudml.org/doc/207783>.

@article{Czornik2006,
abstract = {In this paper we show new formulas for the spectral radius and the spectral subradius of a set of matrices. The advantage of our results is that we express the spectral radius of any set of matrices by the spectral radius of a set of symmetric positive definite matrices. In particular, in one of our formulas the spectral radius is expressed by singular eigenvalues of matrices, whereas in the existing results it is expressed by eigenvalues.},
author = {Czornik, Adam, Jurgas, Piotr},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {symmetric matrices; spectral subradius; spectral radius; singular eigenvalues},
language = {eng},
number = {2},
pages = {183-188},
title = {Some properties of the spectral radius of a set of matrices},
url = {http://eudml.org/doc/207783},
volume = {16},
year = {2006},
}

TY - JOUR
AU - Czornik, Adam
AU - Jurgas, Piotr
TI - Some properties of the spectral radius of a set of matrices
JO - International Journal of Applied Mathematics and Computer Science
PY - 2006
VL - 16
IS - 2
SP - 183
EP - 188
AB - In this paper we show new formulas for the spectral radius and the spectral subradius of a set of matrices. The advantage of our results is that we express the spectral radius of any set of matrices by the spectral radius of a set of symmetric positive definite matrices. In particular, in one of our formulas the spectral radius is expressed by singular eigenvalues of matrices, whereas in the existing results it is expressed by eigenvalues.
LA - eng
KW - symmetric matrices; spectral subradius; spectral radius; singular eigenvalues
UR - http://eudml.org/doc/207783
ER -

References

top
  1. Berger M.A. and Yang Wang (1992): Bounded semigroups ofmatrices. - Lin. Alg. Appl., Vol. 166, No. 1, pp. 21-27. Zbl0818.15006
  2. Czornik A. (2005): On the generalized spectral subradius. - Lin. Alg. Appl., Vol. 407, No. 1, pp. 242-248. Zbl1080.15008
  3. Daubechies I. and Lagarias J.C. (1992a): Two-scale difference equation II. Infinite matrix products, local regularity bounds and fractals. - SIAM J. Math. Anal., Vol. 23, No. 4, pp. 1031-1079. Zbl0788.42013
  4. Daubechies I. and Lagarias J.C (1992b): Sets of matrices all infinite products of which converge. - Linear Alg. Appl., No. 161,pp. 227-263. Zbl0746.15015
  5. Elsner L. (1995): The generalized spectral radius theorem: Ananalytic-geometric proof. - Lin. Alg. Appl., Vol. 220, No. 1, pp. 151-159. Zbl0828.15006
  6. Gol'dsheid I.Ya. and Margulis G.A. (1989): Lyapunov indices of aproduct of random matrices. - Russian Math. Surveys, Vol. 44,No. 1, pp. 11-71. 
  7. Golub G.H. and Loan C.F.V. (1996): Matrix Computations. - 3rd Ed. Baltimore, Johns Hopkins University Press. Zbl0865.65009
  8. Gripenberg G. (1996): Computing the joint spectral radius. - Lin. Alg. Appl., Vol. 234, No. 1, pp. 43-60. Zbl0863.65017
  9. Guglielmi N. and Zennaro M. (2001): On the asymptotic properties of a family of matrices. - Lin. Alg. Appl., Vol. 322,No. 1-3, pp. 169-192. Zbl0971.15016
  10. Gurvits L. (1995): Stability of discrete linear inclusion. - Lin. Alg. Appl., Vol. 231, No. 1, pp. 47-85. Zbl0845.68067
  11. Horn R.A. and Johnson C.R. (1985): Matrix Analysis. - Cambridge, MA: Cambridge Univ. Press. Zbl0576.15001
  12. Jia R.Q. (1995): Subdivision schemes in L_p spaces. - Adv. Comput. Math., Vol. 3, No. 1, pp.309-341. Zbl0833.65148
  13. Michelli C.A. and Prautzsch H. (1989): Uniform refinement of curves. - Lin. Alg. Appl., Vol. 114 and 115, No. 1, pp. 841-870. Zbl0668.65011
  14. Rota G.C. and Strang G. (1960): A note on the joint spectral radius. - Inag. Math. Vol. 22,No. 1, pp. 379-381. Zbl0095.09701
  15. Shih M.H. (1999): Simultaneous Schur stability. - Lin. Alg. Appl., Vol. 287, No. 1-3, pp. 323-336. Zbl0948.15016

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.