Falseness of the Finiteness Property of the Spectral Subradius
International Journal of Applied Mathematics and Computer Science
Analysis of Discrete-Time Linear Switched Systems: A Variational Approach
SIAM Journal on Control and Optimization
Uniform stabilization of discrete-time switched and Markovian jump linear systems
Automatica (Journal of IFAC)
An experimental study of approximation algorithms for the joint spectral radius
Numerical Algorithms
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We prove that there exist (infinitely many) values of the real parameters a and b for which the matrices ( \begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array} \right) \qquad \mbox{ and } \qquad b \left( \begin{array}{ll} 1 & 0 \\ 1 & 1 \end{array} \right) $$ have the following property: all infinite periodic products of the two matrices converge to zero, but there exists a nonperiodic product that doesn't. Our proof is self-contained and fairly elementary; it uses only elementary facts from the theory of formal languages and from linear algebra. It is not constructive in that we do not exhibit any explicit values of a and b with the stated property; the problem of finding explicit matrices with this property remains open.