In recent years this notion has found applications in a large number of engineering fields and is still a topic of active research.
The joint spectral radius was introduced in 1960 by Gian-Carlo Rota and Gilbert Strang,[1] two mathematicians from MIT, but started attracting attention with the work of Ingrid Daubechies and Jeffrey Lagarias.
[2] They showed that the joint spectral radius can be used to describe smoothness properties of certain wavelet functions.
It is known that the joint spectral radius quantity is NP-hard to compute or to approximate, even when the set
[5] 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.
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.
[6] 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.
The second family of methods approximate the extremal norm with modern optimization techniques, such as ellipsoid norm approximation,[9] semidefinite programming,[10][11] Sum Of Squares,[12] and conic programming.
[13] 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.
Related to the computability of the joint spectral radius is the following conjecture:[14] "For any finite set of matrices
[19][20] The joint spectral radius was introduced for its interpretation as a stability condition for discrete-time switching dynamical systems.
The joint spectral radius became popular when Ingrid Daubechies and Jeffrey Lagarias showed that it rules the continuity of certain wavelet functions.
However, many 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
The Lyapunov exponent of the set of matrices characterizes the rate of growth of the geometric average.