= Exponential-surjectivity obstruction from distinct negative eigenvalues
A real matrix with at least one simple negative <eigenvalue> cannot be a <matrix exponential> of a real matrix: a logarithm would commute with it and preserve each one-dimensional real negative-eigenvalue space, where exponentiation can only produce a positive <eigenvalue>. In particular $\operatorname{diag}(-2,-1/2)$ belongs to the connected <special linear group> $\mathrm{SL}_2(\mathbb R)$ but has no real logarithm.
Back to article page
