联合谱半径

联合谱半径（joint spectral radius）为一数学名词，是将传统上针对矩阵的谱半径表示法，扩展到矩阵集合的表示法。近年来此表示法已应用在许多工程领域中，也是目前研究的热门主题。

概述[编辑]

矩阵集合的联合谱半径是在集合中矩阵乘积的最大渐近成长率。针对有限集合（或是更广义的紧凑集合） ${\mathcal {M}}=\{A_{1},\dots ,A_{m}\}\subset \mathbb {R} ^{n\times n},$ ，其联合谱半径定义如下：

\rho ({\mathcal {M}})=\lim _{k\to \infty }\max {\{\|A_{i_{1}}\cdots A_{i_{k}}\|^{1/k}:A_{i}\in {\mathcal {M}}\}}.\,

可以证明其极限存在，而且其数值不会随所选择的矩阵范数种类而改变（这对任何矩阵范数都成立，不过若矩阵范数有次可乘性sub-multiplicative，更容易证明）。联合谱半径的概念是在1960年由麻省理工学院的两位数学家吉安-卡洛·罗塔及威廉·吉尔伯特·斯特朗发明^[1]，不过在英格丽·多贝西及杰佛瑞·拉加里亚斯（英语：Jeffrey Lagarias）的研究后，才开始受到注意^[2]，他们证明了联合谱半径可以用来描述特定小波函数的光滑性^[3]。之后就提出了许多相关的应用。目前已知联合谱半径的量值求值，不论是要计算或只是近似，在运算复杂度上都是NP困难，就算集合 ${\mathcal {M}}$ 中只有二个矩阵，其中所有非零元素都相同也是一样^[4]。而且， $\rho \leq 1?$ 这个问题是不可判定问题^[5]。不过，近年来已对此问题有多一些的了解，似乎在实务上，可以计算联合谱半径到令人满意的精度，而且对于一些工程及数学问题，可以有一些有趣的洞察。

计算[编辑]

近似算法[编辑]

虽然在联合谱半径的可计算性理论上有一些负面的结果，不过已有提出一些在实务上可以良好运作的方法。目前已找到算法，可以达到任意的精度，所需要的时间也是事先可以计算得知。这类的算法可以视为是近似向量范数（称为极值范数extremal norm）中的单位球^[6]。一般会将算法分为两类：第一类是多义范数法（polytope norm method），透过计算点的长轨迹来建构极值范数^[7]^[8]，此方法的好处是在最理想的情形下，此方法可以找到联合谱半径的精确值，而且可以证明这个值就是正确值。

第二种方式是用“近代最佳化技巧”（modern optimization techniques）来近似极值范数，例如椭圆范数近似（ellipsoid norm approximation）^[9]、半正定规划^[10]^[11]、多项式平方和（英语：Polynomial SOS）^[12]、圆锥规划（英语：Conic optimization）^[13]。这些方法的好处是容易实现，而且实务上此方式所产生的联合谱半径，一般来说是在最理想的范围内。

有限性猜想[编辑]

有关联合谱半径的可计算性，存在以下的猜想^[14]：

“针对任何有限个的矩阵集合 ${\mathcal {M}}\subset \mathbb {R} ^{n\times n},$ ，存在一个矩阵乘积 $A_{1}\dots A_{t}$ 使得

\rho ({\mathcal {M}})=\rho (A_{1}\dots A_{t})^{1/t}.

”

上式中的“ $\rho (A_{1}\dots A_{t})$ ”是指矩阵 $A_{1}\dots A_{t}$ 在传统意义下的谱半径。

此猜想在1995年提出，在2003年证否^[15]。在参考资料中的反例用到了进阶的量度理论（measure-theoretical）概念。之后，也找到了许多的反例，包括只用到简单组合数学性质的矩阵^[16]以及另一个用到动态系统概念的反例^[17]。近来也提出了一显式的反例^[18]。许多相关的问题还没有证明，例如对于成对的逻辑矩阵，此猜想是否成立^[19]^[20]。

应用[编辑]

联合谱半径的出现，是为了作为离散时间切换动力系统的稳定性条件。而以下方程定义的系统

x_{t+1}=A_{t}x_{t},\quad A_{t}\in {\mathcal {M}}\,\forall t

为李雅普诺夫稳定性若而唯若 $\rho ({\mathcal {M}})<1.$ 。

因为英格丽·多贝西及杰佛瑞·拉加里亚斯将联合谱半径应用在小波函数的连续性上，因此联合谱半径受到许多人的注意。之后的应用包括有数论、信息理论、自治代理（英语：autonomous agent）共识、字的组合数学（英语：combinatorics on words）等。

参考资料[编辑]

^ G. C. Rota and G. Strang. "A note on the joint spectral radius." Proceedings of the Netherlands Academy, 22:379–381, 1960. [1]
^ 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.
^ 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.
^ 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.
^ 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.
^ N. Barabanov. "Lyapunov indicators of discrete inclusions i–iii." Automation and Remote Control, 49:152–157, 283–287, 558–565, 1988.
^ V. Y. Protasov. "The joint spectral radius and invariant sets of linear operators." Fundamentalnaya i prikladnaya matematika, 2(1):205–231, 1996.
^ 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.
^ 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.
^ T. Ando and M.-H. Shih. "Simultaneous contractibility." SIAM Journal on Matrix Analysis and Applications, 19(2):487–498, 1998.
^ V. D. Blondel and Y. Nesterov. "Computationally efficient approximations of the joint spectral radius." SIAM Journal of Matrix Analysis, 27(1):256–272, 2005.
^ 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.
^ 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.
^ 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.
^ T. Bousch and J. Mairesse. "Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture." Journal of the American Mathematical Society, 15(1):77–111, 2002.
^ 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.
^ V. Kozyakin Structure of Extremal Trajectories of Discrete Linear Systems and the Finiteness Conjecture, Automat. Remote Control, 68 (2007), no. 1, 174–209/
^ 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.
^ A. Cicone, N. Guglielmi, S. Serra Capizzano, and M. Zennaro. "Finiteness property of pairs of 2 × 2 sign-matrices via real extremal polytope norms." Linear Algebra and its Applications, 2010.
^ R. M. Jungers and V. D. Blondel. "On the finiteness property for rational matrices." Linear Algebra and its Applications, 428(10):2283–2295, 2008.

延伸阅读[编辑]

Raphael M. Jungers. The joint spectral radius, Theory and applications. Springer. 2009. ISBN 978-3-540-95979-3.

Vincent D. Blondel; Michael Karow; Vladimir Protassov; Fabian R. Wirth (编). Linear Algebra and its Applications: special issue on the joint spectral radius 428. Elsevier. 2008. |journal=被忽略 (帮助); |issue=被忽略 (帮助)

Antonio Cicone. PhD thesis. Spectral Properties of Families of Matrices. Part III (PDF). 2011 [2020-01-28]. （原始内容 (PDF)存档于2012-03-31）.

Jacques Theys. PhD thesis. Joint Spectral Radius: Theory and approximations. (PDF). 2005 [2020-01-28]. （原始内容 (PDF)存档于2007-06-13）.

[1] G. C. Rota and G. Strang. "A note on the joint spectral radius." Proceedings of the Netherlands Academy, 22:379–381, 1960. [1]

[2] 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.

[3] 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.

[4] 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.

[5] 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.

[6] N. Barabanov. "Lyapunov indicators of discrete inclusions i–iii." Automation and Remote Control, 49:152–157, 283–287, 558–565, 1988.

[7] V. Y. Protasov. "The joint spectral radius and invariant sets of linear operators." Fundamentalnaya i prikladnaya matematika, 2(1):205–231, 1996.

[8] 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.

[9] 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.

[10] T. Ando and M.-H. Shih. "Simultaneous contractibility." SIAM Journal on Matrix Analysis and Applications, 19(2):487–498, 1998.

[11] V. D. Blondel and Y. Nesterov. "Computationally efficient approximations of the joint spectral radius." SIAM Journal of Matrix Analysis, 27(1):256–272, 2005.

[12] 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.

[13] 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.

[14] 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.

[15] T. Bousch and J. Mairesse. "Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture." Journal of the American Mathematical Society, 15(1):77–111, 2002.

[16] 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.

[17] V. Kozyakin Structure of Extremal Trajectories of Discrete Linear Systems and the Finiteness Conjecture, Automat. Remote Control, 68 (2007), no. 1, 174–209/

[18] 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.

[19] A. Cicone, N. Guglielmi, S. Serra Capizzano, and M. Zennaro. "Finiteness property of pairs of 2 × 2 sign-matrices via real extremal polytope norms." Linear Algebra and its Applications, 2010.

[20] R. M. Jungers and V. D. Blondel. "On the finiteness property for rational matrices." Linear Algebra and its Applications, 428(10):2283–2295, 2008.

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