Problem of Finiteness Conjecture and Joint Spectral Radius


This paper studies the constraint of a vector norm under periodic and aperiodic action of matrices from a finite set of matrices with rational elements as well as the presence of AZR and PZR. We consider a complex case when every matrix from has its own numbers that are both bigger and smaller units. By solving the problem whether has characteristics of AZR and PZR, we investigate a lower spectral radius (LSR) for such set The conducted research proves that and are absent for the finite set of matrices which complies with specific conditions. We also determine that We prove that AZR and PZR conditions for and vectors from are fulfilled. Finally, we determine that the systems of matrices are present above for which ES occurs.

Publication year: 
С. 88–92., укр., Бібліогр.: 4 назви.

1. Blondel V.D., Nesterov Yu., Theys J. Computing the Joint Spectral Radius of a Set of Matrices // 23rd Benelux Meeting on Systems and Control, Helvoirt. The Netherlands, paper FrP06-3. — March 17—19. — 2004. —P. 103.
2. Blondel V.D., Gaubert S., and Tsitsiklis J.N. Approximating the Spectral Radius of Sets of Matrices in the Max-Algebra is NP-hard // IEEE Transactions on Automatic Control. — 2000. — P. 1762—1765.
3. Theys Jacques. Joint Spectral Radius: Theory and Approximations. Ph. thesis in Combinatorics and Optimization. Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Docteur en Sciences Appliques. — 2005. — 198 p.
4. Blondel V.D., Theys J. and Vladimirov A.A. An Elementary Counterexample to the Finiteness Conjecture // SIAM J. on Matrix Analysis. — 2003. — 24, N 4. — P. 963—970.

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