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| In mathematics, the '''joint spectral radius''' is a generalization of the classical notion of [[spectral radius]] of a matrix, to sets of matrices. In recent years this notion has found applications in a large number of engineering fields and is still a topic of active research.
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| ==General description==
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| The joint spectral radius of a set of matrices is the maximal asymptotic growth rate of products of matrices taken in that set. For a finite (or more generally compact) set of matrices <math>\mathcal M=\{A_1,\dots, A_m\} \subset \mathbb R^{n \times n},</math> the joint spectral radius is defined as follows:
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| : <math>\rho (\mathcal M)= \lim_{k \to
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| \infty}\max{\{ \|A_{i_1}\cdots A_{i_k}\|^{1/k}:A_i\in\mathcal M\}}. \, </math>
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| It can be proved that the limit exists and that the quantity actually does not depend on the chosen matrix norm (this is true for any norm but particularly easy to see if the norm is [[Matrix norm|sub-multiplicative]]). The joint spectral radius was introduced in 1960 by [[Gian-Carlo Rota|Rota]] and [[Gilbert Strang|Strang]],<ref>G. C. Rota and G. Strang. "A note on the joint spectral radius." Proceedings of the Netherlands Academy, 22:379–381, 1960. [http://books.google.it/books?id=x7zKLGu3g9IC&lpg=RA1-PA40&ots=rxuCbvNTxI&dq=a%20note%20on%20the%20joint%20spectral%20radius%20rota%20strang&lr&pg=PA74#v=onepage&q&f=false]</ref> two mathematicians from [[Massachusetts Institute of Technology|MIT]], but started attracting attention with the work of [[Ingrid Daubechies]] and [[Jeffrey Lagarias]].<ref>Vincent D. Blondel. The birth of the joint spectral radius: an interview with Gilbert Strang. Linear Algebra and its Applications, 428:10, pp. 2261–2264, 2008.</ref> They showed that the joint spectral radius can be used to describe smoothness properties of certain [[Wavelet#Wavelet function|wavelet functions]].<ref>I. Daubechies and J. C. Lagarias. "Two-scale difference equations. ii. local regularity, infinite products of matrices and fractals." SIAM Journal of Mathematical Analysis, 23, pp. 1031–1079, 1992.</ref> A wide number of applications have been proposed since then. It is known that the joint spectral radius quantity is [[NP-hard]] to compute or to approximate, even when the set <math>\mathcal M</math> consists of only two matrices with all nonzero entries of the two
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| matrices which are constrained to be equal.<ref>J. N. Tsitsiklis and V. D. Blondel. "Lyapunov Exponents of Pairs of Matrices, a Correction." ''[[Mathematics of Control, Signals, and Systems]]'', 10, p. 381, 1997.</ref> Moreover, the question "<math>\rho\leq 1 ?</math>" is an [[undecidable problem]].<ref>Vincent D. Blondel, John N. Tsitsiklis. "The boundedness of all products of a pair of matrices is undecidable." Systems and Control Letters, 41:2, pp. 135–140, 2000.</ref> Nevertheless, in recent years much progress has been done on its understanding, and it appears that in practice the joint spectral radius can often be computed to satisfactory precision, and that it moreover can bring interesting insight in engineering and mathematical problems.
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| ==Computation==
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| ===Approximation algorithms===
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| In spite of the negative theoretical results on the joint spectral radius computability, methods have been proposed that perform well in practice. Algorithms are even known, which can reach an arbitrary accuracy in an a priori computable amount of time. These algorithms can be seen as trying to approximate the unit ball of a particular vector norm, called the extremal norm.<ref>N. Barabanov. "Lyapunov indicators of discrete inclusions i–iii." Automation and Remote Control, 49:152–157, 283–287, 558–565, 1988.</ref> One generally distinguishes between two families of such algorithms: the first family, called '''polytope norm methods''', construct the extremal norm by computing long trajectories of points.<ref>V. Y. Protasov. "The joint spectral radius and invariant sets of linear operators." Fundamentalnaya i prikladnaya matematika, 2(1):205–231, 1996.</ref><ref>N. Guglielmi, F. Wirth, and M. Zennaro. "Complex polytope extremality results for families of matrices." SIAM Journal on Matrix Analysis and Applications, 27(3):721–743, 2005.</ref> An advantage of these methods is that in the favorable cases it can find the exact value of the joint spectral radius and provide a certificate that this is the exact value.
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| The second methods approximate the extremal norm with '''modern optimization techniques''', like ellipsoid norm approximation,<ref>Vincent D. Blondel, Yurii Nesterov and Jacques Theys, On the accuracy of the ellipsoid norm approximation of the joint spectral radius, Linear Algebra and its Applications, 394:1, pp. 91–107, 2005.</ref> [[semidefinite programming]],<ref>T. Ando and M.-H. Shih. "Simultaneous contractibility." SIAM Journal on Matrix Analysis and Applications, 19(2):487–498, 1998.</ref><ref>V. D. Blondel and Y. Nesterov. "Computationally efficient approximations of the joint spectral radius." SIAM Journal of Matrix Analysis, 27(1):256–272, 2005.</ref> [[Polynomial SOS|Sum Of Squares]],<ref>P. Parrilo and A. Jadbabaie. "Approximation of the joint spectral radius using sum of squares." Linear Algebra and its Applications, 428(10):2385–2402, 2008.</ref> [[Conic optimization|conic programming]].<ref>V. Protasov, R. M. Jungers, and V. D. Blondel. "Joint spectral characteristics of matrices: a conic programming approach." SIAM Journal on Matrix Analysis and Applications, 2008.</ref> The advantage of these methods is that they are easy to implement, and in practice, they provide in general the best bounds on the joint spectral radius. | |
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| ===The finiteness conjecture===
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| Related to the computability of the joint spectral radius is the following conjecture:<ref>J. C. Lagarias and Y. Wang. "The finiteness conjecture for the generalized spectral radius of a set of matrices." Linear Algebra and its Applications, 214:17–42, 1995.</ref>
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| "For any finite set of matrices <math>\mathcal M \subset \mathbb R^{n \times n},</math> there is a product <math> A_1\dots A_t</math> of matrices in this set such that
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| :<math>\rho(\mathcal M) = \rho(A_1 \dots A_t)^{1/t}.</math>"
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| In the above equation "<math> \rho(A_1 \dots A_t)</math>" refers to the classical [[spectral radius]] of the matrix <math>A_1 \dots A_t.</math>
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| This conjecture, proposed in 1995, has been proved to be false in 2003,.<ref>T. Bousch and J. Mairesse. "Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture." Journal of the Mathematical American Society, 15(1):77–111, 2002.</ref> The counterexample provided in that reference uses advanced measure-theoretical ideas. Subsequently, many other counterexamples have been provided, including an elementary counterexample that uses simple combinatorial properties matrices <ref>V. D. Blondel, J. Theys and A. A. Vladimirov, An elementary counterexample to the finiteness conjecture, SIAM Journal on Matrix Analysis, 24:4, pp. 963–970, 2003.</ref> and a counterexample based on dynamical systems properties.<ref>V. Kozyakin
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| Structure of Extremal Trajectories of Discrete Linear Systems and the Finiteness Conjecture, Automat. Remote Control, 68 (2007), no. 1, 174–209/</ref> Recently an explicit counterexample has been proposed in.<ref>Kevin G. Hare, Ian D. Morris, Nikita Sidorov, Jacques Theys. An explicit counterexample to the Lagarias–Wang finiteness conjecture, Advances in Mathematics, 226, pp. 4667-4701, 2011.</ref> Many questions related to this conjecture are still open, as for instance the question of knowing whether it holds for pairs of [[Binary matrix|binary matrices]].<ref>A. Cicone, N. Guglielmi, S. Serra Capizzano, and M. Zennaro. "Finiteness property of pairs of
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| 2 × 2 sign-matrices via real extremal polytope norms." Linear Algebra and its Applications, 2010.</ref><ref>R. M. Jungers and V. D. Blondel. "On the finiteness property for rational matrices." Linear Algebra and its Applications, 428(10):2283–2295, 2008.</ref>
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| ==Applications==
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| The joint spectral radius was introduced for its interpretation as a stability condition for discrete-time switching [[dynamical system]]s. Indeed, the system defined by the equations
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| :<math>x_{t+1}=A_tx_{t}, \quad A_t\in \mathcal M \, \forall t</math>
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| is [[Lyapunov stability|stable]] if and only if <math>\rho(\mathcal M)<1.</math>
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| The joint spectral radius became popular when [[Ingrid Daubechies]] and [[Jeffrey Lagarias]] showed that it rules the continuity of certain wavelet functions. Since then, it has found many applications, ranging from number theory to information theory, [[autonomous agent]]s consensus, [[combinatorics on words]],...
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| ==Related notions==
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| The joint spectral radius is the generalization of the [[spectral radius]] of a matrix for a set of several matrices. However, much more quantities can be defined when considering a set of matrices: The '''joint spectral subradius''' characterizes the minimal rate of growth of products in the semigroup generated by <math>\mathcal M</math>.
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| The '''p-radius''' characterizes the rate of growth of the <math>L_p</math> average of the norms of the products in the semigroup.
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| The '''Lyapunov exponent''' of the set of matrices characterizes the rate of growth of the geometric average.
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| == Further reading ==
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| *{{cite book
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| | author = Raphael M. Jungers
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| | year = 2009
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| | title = The joint spectral radius, Theory and applications
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| | publisher = Springer
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| | isbn = 978-3-540-95979-3
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| }}
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| *{{cite book
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| | editor = Vincent D. Blondel, Michael Karow, Vladimir Protassov, and Fabian R. Wirth
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| | year = 2008
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| | journal = Linear Algebra and its Applications
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| | title = Linear Algebra and its Applications: special issue on the joint spectral radius
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| | publisher = Elsevier
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| | volume = 428
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| | issue = 10
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| }}
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| *{{cite web
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| |url=http://www.math.msu.edu/~cicone/papers/AntonioCiconeThesis.pdf
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| |title=PhD thesis. Spectral Properties of Families of Matrices. Part III
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| |author=Antonio Cicone
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| |year=2011
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| }}
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| *{{cite web
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| |url=http://www.inma.ucl.ac.be/~blondel/05thesetheys.pdf
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| |title=PhD thesis. Joint Spectral Radius: Theory and approximations.
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| |author=Jacques Theys
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| |year=2005
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| }}
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| == References ==
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| {{Reflist}}
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| <!--- Categories --->
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| [[Category:Control theory]]
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| [[Category:Linear algebra]]
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