# Explanation We would like a geometric interpretation of the [[trace]] of a generator, i.e., an element of a [[Lie Algebra]]. Let $X \in \mathfrak{gl}(n,\mathbb{C})$, i.e., $X$ is an element of the [[Lie Algebra]] of the [[general linear group]]. Define an arbitrary [[One-Parameter Subgroup]] $\{e^{tX}\} \subset GL(n, \mathbb{C})$, then we have: $ \det(e^{tX}) = e^{\text{tr}(tX)} = e^{t \text{Tr} X} $ where we have used the properties of the [[determinant]] to get the second equality. Taking the derivative with respect to $t$ and evaluating at $t=0$, we get: $ \frac{d}{dt}\det(e^{tX})\bigg|_{t=0} = \text{tr}(X) $ Since the determinant measures how an operator (or matrix) changes volumes, this equality says that the trace of a generator gives the *rate* at which volume change under the action of the corresponding one-parameter subgroup $e^{tX}$.