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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. ==General description== 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 the joint spectral radius is defined as follows: : 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 sub-multiplicative). The joint spectral radius was introduced in 1960 by Gian-Carlo Rota and Gilbert Strang,〔G. C. Rota and G. Strang. "A note on the joint spectral radius." Proceedings of the Netherlands Academy, 22:379–381, 1960. ()〕 two mathematicians from MIT, but started attracting attention with the work of Ingrid Daubechies and Jeffrey Lagarias.〔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.〕 They showed that the joint spectral radius can be used to describe smoothness properties of certain wavelet functions.〔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.〕 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 consists of only two matrices with all nonzero entries of the two matrices which are constrained to be equal.〔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.〕 Moreover, the question "" is an undecidable problem.〔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.〕 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Joint spectral radius」の詳細全文を読む スポンサード リンク
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