Exponential parametrization for diagonal SL(2,R)

Question

I already saw this question (and the linked questions) but mine is slightly different. I am asked to prove the fact that the exponential parameterization isn't a good one for $SL(2,\mathbb{R})$ using some diagonal matrices as examples. I'm aware that one can find some (upper triangular) matrices that don't have a bijective exponential parameterization, but I'm pretty sure that any $SL$ matrices of the form $diag(x,x^{-1})$ can be written as $diag(e^a,e^{-a})$. Am I wrong, or is the question just not answerable using diagonal matrices?

edit: thanks to Moishe Kohan, I noticed that there are some $x$ so that $diag(x,x^{-1})$ can't be written as $diag(e^a,e^{-a})$ - the negatives. But I also noticed that a diagonal matrix written as an exponential of another matrix $diag(x, x^{-1})=e^A$ doesn't imply that A is diagonal! So this still doesn't answer the question: can all diagonal $SL$ matrices be written with an exponential parameterization?

@MoisheKohan that was one of the possibilities. This is why I wrote "am I wrong, or..." — Mauro Giliberti, Jan 08 '21 at 17:22
Yes, so the hint is that you are wrong, and you should look closer. Start by answering the question "which nonzero real numbers have the form $e^t$, for some $t\in {\mathbb R}$." — Moishe Kohan, Jan 08 '21 at 17:24

score 1 · Accepted Answer · answered Jan 08 '21 at 18:13

1

Hints:

Suppose that $D$ is a diagonal matrix and $D=\exp(A)$. Relate the eigenvalues of $D$ and of $A$. (If this is too difficult, assume that $D$ has distinct diagonal entries.)
Consider what happens when $D$ has negative diagonal entries.

answered Jan 08 '21 at 18:13

Moishe Kohan

111,854

Exponential parametrization for diagonal SL(2,R)

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