Can anyone explain to me the meaning of the statement below from Tu - Introduction to Manifolds (specifically section 15.3).
To compute the differential of a map on a subgroup of $GL(n,R)$, we need a curve of nonsingular matrices. Because the matrix exponential is always nonsingular, it is uniquely suited for this purpose.
Why is the exponential matrix so important? Is it because it represents some sort of parameterization?
No. In this context it is imprtant because, for any square matrix $A$, $e^A$ is nonsingular, since$$\det(e^A)=e^{\operatorname{tr}A}\neq0.$$