Joint spectral radius

The spectral radius of a square matrix is the absolute value of its largest eigenvalue. 

The joint spectral radius of two square matrices A and B of the same size, is defined by the following steps  

We can get some handle on this value by noting that any norm is an upper bound on the spectral radius of a matrix. If, during the tending of to to oo, it is found that the matrix given by one of the product sequences has a norm equal to its spectral radius, then that will be the joint spectral radius of the A and B. 

The loo norm of such a matrix is much cheaper to compute than the eigenvalues, and as to increases the nth root of the lowest norm so far of the product sequences converges to the joint spectral radius, as does (by definition) the nth root of the largest eigenvalue so far. 

Other useful properties are that if A and B share an eigenvector, then both AB and BA will also share that eigenvector, and the corresponding eigenvalue of the product will just be the product of the eigenvalues of A and B. The square root cannot be larger than the larger of these factors. 

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Contribution to the joint spectral radius from a shared... 

In particular, the joint spectral radius of any matrix with itself is just the spectral radius of that matrix. 

Yet another twist shows that both of these two computations can be understood in terms of a standard property of a pair of matrices, their Joint Spectral Radius. 

This is what the joint spectral radius analysis does. The joint spectral radius of the two matrices L and R, is defined as the limit, as n tends to oo, of the value of the nth root of the largest dominant eigenvalue of any of the matrices formed by taking all possible product sequences of length n of L and R. 

The key theorem is that upper bounds, given by norms, and lower bounds, given by actually taking eigenvalues as per the definition of joint spectral radius, do in fact converge to the same value. 

Unfortunately there is no simple method of computing directly the joint spectral radius for two given general matrices. The best we can do is to compute the upper and lower bounds for larger and larger values of n and watch them converge. 

The subtlety of the joint spectral radius results is that the fact that both the highest eigenvalue and the lowest Zoo norm converge to a well defined joint spectral radius value means that the Zoo norm of any finite power of the kernel gives an upper bound on the eigenvalues of all powers of the kernel, and therefore a lower bound on any constructible discontinuity. We do not have to prove that the Ly norm shrinks by the arity every time we add an extra (1 — za)/a( 1 — z) factor. 

It is of interest to note that schemes containing the kernel [1,0,1], at c = 2, are in fact unnormalised schemes. Their continuity is exactly that determined by the joint spectral radius analysis of the kernel without any need for normalisation. 

All product sequences of A and B of length l will therefore share the eigenvector V, with an eigenvector which is a weighted geometric mean of Xla and Aj, which will not exceed the larger of Xla and A. The joint spectral radius of A and B will not, therefore, come from any mixed sequence of A and B. 

Thus the joint spectral radius of A and B will either be the larger of the spectral radii of A and B, or else come from the part of the spectrum which is not associated with the nested invariant subspaces. 

The joint spectral radius approach appeared in a paper by Ingrid Dau-bechies and Jeffrey Lagarias[DaLa91] in 1991, and also in one by Hartmut Prautzsch and Charles Micchelli[PM87] in 1987. Efficient computation is still a hot topic. A strong competitor for the ideas described in section 18.3 above is the depth first search method developed by the team of Ulrich Reif. 

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