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.


Answers 1

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
loup blanc
June 13, 2019 12:01 PM

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