WDS wrote: << Now wants, given any finite set of matrices, to prove any product of n of the matrices has all eigenvalues e obeying |e|<c^n for some positive constant c. (Is that right? That is now your goal?) >> Not quite --- c was to be chosen in advance (and sharp). << The spectral norm and the Frobenius norm of the matrices are submultiplicative (so both should work), and the max |eigenvalue| is upper bounded by sqrt(FrobeniusNorm). >> Aha --- that's a good point about the spectral norm. In the example I gave, c is in fact the spectral norm of each factor. If it turns out to be the spectral norm of all the generators, then I'm home and dry! On 3/21/14, Victor S. Miller <victorsmiller@gmail.com> wrote:
The rate of growth of a matrix semigroup is often quite delicate. See http://www.ams.org/journals/mcom/2000-69-231/S0025-5718-99-01145-X/S0025-571... for a fun example.
Victor
Nice paper, but with woeful implications if Warren's idea doesn't do the trick. Random matrix theory is one of Terry Tao's hobbies --- 'nuff said? WFL
On Mar 21, 2014, at 14:35, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Sorry everybody --- I fouled up by oversimplifying the problem.
So let's make it a semigroup generated by a given finite set of integer matrices (it seems experimentally that my constraints can be jettisoned), and I'm trying to show that every product of s generators has all eigenvalues bounded in modulus by c^s (where c is also given).
A toy example, which I can actually decide by elementary means, is generated by 10 2x2 matrices shown below, where c = (1 + rt5)/2 is the golden section.
Notice that the product
[1 0] [1 1] = [1 1] [1 1] [0 1] = [1 2]
has eigenvalue c^2 , despite both generators having unit eigenvalues.
WFL
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matOOOO; [1 1] [1 0] matOOOI; [1 0] [1 0] matOOII; [1 0] [1 1] matOIOO; [1 1] [0 0] matOIOI; [1 0] [0 1] matOIIO; [0 1] [1 0] matOIII; [0 0] [1 1] matIIOO; [1 1] [0 1] matIIOI; [0 1] [0 1] matIIII; [0 1] [1 1]