Existence of a logarithm for every invertible complex matrix
= Existence of a logarithm for every invertible complex matrix
An invertible complex <Jordan block> $\lambda I+N$, with $N^s=0$, has logarithm $\ell I+\sum_{k=1}^{s-1}(-1)^{k+1}(N/\lambda)^k/k$, where $e^\ell=\lambda$. Applying this construction blockwise and conjugating back proves surjectivity of the <matrix exponential> on the complex <general linear group>. There is no globally single-valued continuous choice of logarithm.