Sunday, 12 May 2013

matrices - Radical of mathfrakgln

I find it intuitive enough that the radical of glnF is the scalar matrices, but I have trouble finding an easy, but complete proof:




Proof. Let s denote the scalar matrices. Clearly srad(glnF). Suppose that rad(glnF) is generated by more than one element, so that Xrad(glnF), but Xs. We can change basis such that rad(glnF) is upper-triangular. (The change of basis leaves the scalar matrices invariant.) Then X is upper-triangular.



I want to conclude that there exists a YglnF s.t. [X,Y] is not upper-triangular. How can I see that Y always exists, except by waving hands?

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