# Burnside's theorem on invariant subpaces

by PainBouchon   Last Updated June 13, 2019 12:20 PM

One version of this theorem states that if $$E$$ is a complex vector space with $$dim(E)<+\infty$$, and $$A$$ an unitary sub-algebra of $$\mathcal{L}(E)$$ for which there are no non-trivial subspaces $$F$$ invariant by all the elements of $$A$$ (simultaneously), then $$A=\mathcal{L}(E)$$. In other words, if $$A$$ is a strict unitary subalgebra of $$\mathcal{L}(E)$$, then there is a subspace $$F$$ non-trivial invariant by all elements of A.

Do you know some simple examples of such $$A$$ ? Could we prove some results concerning simultaneous triangularization using directly this theorem? I have only found other theorems resulting of this one, or examples not very simple.

Thanks!

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Two examples.

i) Two randomly chosen complex $$n\times n$$ matrices $$A,B$$ span the full matrix algebra with probability $$1$$. cf. my post in

Probability that two random matrices span the full matrix algebra

Note that it's not obvious...

ii) -Easier- Let $$A,B$$ be two $$2\times 2$$ matrices s.t. $$e^Ae^B=e^{A+B}\not= e^Be^A$$. Show that $$A,B$$ are simultaneously triangularizable.

loup blanc
June 13, 2019 12:01 PM

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