Prove that $\operatorname{SL}(n,\Bbb R)$ is connected using the decomposition $\exp(A)\exp(S)$

Question

Using this fact:

Every $X \in \operatorname{SL}(n,\Bbb R)$ can be decomposed as $X=\exp(A)\exp(S)$, with $A, B \in M_n(\Bbb R), \operatorname{tr}(A)=\operatorname{tr}(B)=0$ (i.e $A,B \in \mathfrak{sl}(n,\mathbb R)$) $A$ skew-symmetric, $S$ symmetric

How can I prove that $\operatorname{SL}(n,\Bbb R)$ is connected?

Answer 1

Hint: function $t \mapsto \exp((1-t)S)\exp((1-t)A)\quad\quad$ gives a path from $\exp(S)\exp(A)\quad$ to the identity matrix $I$. Use this to show that $\mathrm{SL}(n, \Bbb{R})$ is path-connected and hence connected.