On the joint spectral radius of matrices of order 2 with equal spectral radius
Many applications like stability analysis of stochastic dynamical systems, or in approximation theory use the joint spectral radius to verify the convergence or smoothness of subdivision algorithms. The joint spectral radius of two matrices \(A_1,A_2\in \mathbb{R}^{d\times d}\) is \(\mathrm{jsr}(A_1,A_2):=\lim_{n \to \infty}\max \{\| T_1T_2\cdots T_n\|^{1/n} \; | \; T_i=A_1 \text{ or } A_2 \}\) and it measures the growth (or decrease) rate of products of matrices taken from a set. In the Introduction of this paper, an accurate presentation of the problem of computing spectral radius and of the literature around this subject is given. The author shows the difficulty of determining the joint spectral radius and the few results that were produced in this direction starting from 1990 by Kozyakin that proved that \(\{(A,B)\in (\mathbb{R}^{2\times 2})^2 \; | \; \mathrm{jsr}(A,B)<1\}\) is not a semi-algebraic set. Some of the known results are about the polynomial-time approximation algorithms for the joint spectral radius (\textit{V. D. Blondel} and \textit{J. N. Tsitsiklis} showed in 1997 that until \(P=NP\) there is no such an approximation algorithm [Math. Control Signals Syst. 10, No.~1, 31--40 (1997; Zbl 0888.65044)]). The joint spectral radius seems to be known in the case of pairs of matrices of order 2 both when one has negative determinant [the author and \textit{X. Zhou}, SIAM J. Matrix Anal. Appl. 22, No.~1, 242--257 (2000; Zbl 0964.42017)] and when of the matrix is singular [\textit{R. Lima} and \textit{M. Rahibe}, J. Phys. A, Math. Gen. 27, No.~10, 3427--3437 (1994; Zbl 0830.15018)]. Other examples were given by \textit{T. N. T. Goodman, C. A. Micchelli} and \textit{J. D. Ward} [in: Recent advances in wavelet analysis. Boston, MA: Academic Press, Inc. Wavelet Anal. Appl. 3, 335--360 (1994; Zbl 0840.42020)] and by \textit{L. Villemoes} [SIAM J. Math. Anal. 25, No.~5, 1433--1460 (1994; Zbl 0809.42016)]. The main result of the paper under review is the computation of two palindromic matrices \(L,P\) of order 2. It is proved that \(\mathrm{jsr}(L,R)=\max (\rho (L), \sqrt{\rho(L,R)})\) where \(\rho(A)\) is the spectral radius of the matrix (A): \(\rho(A):=\max \{|\lambda| \, : \, \lambda \text{ is an eigenvalue of } A\}\). Moreover, the technique of the proof of this result is used by the author to generalize the computation of the spectral radius the similar matrices with certain relation between the entries. At the end of the paper, some examples and an application of the joint spectral radius formula on a subdivision scheme is given.
scientific article; zbMATH DE number 5760442
scientific article