# 联合谱半径

## 概述

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

## 计算

### 有限性猜想

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

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

## 应用

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

## 相关的表示法

• 联合谱次幅（joint spectral subradius）表示由${\displaystyle {\mathcal {M}}}$产生的半群最小成长速率乘积。
• p-半径（p-radius）表示此半群内乘积范数之${\displaystyle L_{p}}$平均的成长速率。
• 矩阵集合的李亚普诺夫指数（Lyapunov exponent）表示其几何平均的成长速率。

## 参考资料

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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.
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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.
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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.
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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.

## 延伸阅读

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