Mathematics Colloquia and Seminars

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In our upper division calculus we learned how to generalize the notion of derivative. We could then speak of the derivative—sometimes called the Frechet-derivative—of maps not just from R to R but from Rⁿ to R^m. In fact, with Real Analysis under our belts, it becomes a triviality to generalize further to maps $f: X --> Y$, where $X, Y$ are arbitrary normed F-vector spaces for F = R or F = C.

In this talk the first thing we will do is make such a more general definition. We will then illustrate it in action by showing that the familiar determinant function from $n$ by $n$ complex matrices to complex numbers, is Frechet-differentiable, and by providing an explicit formula for its derivative. The proof proves to be an interesting mixture of linear algebra and analysis. Given how ubiquitous the determinant function is, there are no doubt many applications of this result. In particular, we will conclude the talk by showing that the inversion map $A --> A^(-1)$ on GL(n,C) is Frechet-differentiable by providing an explicit formula for its derivative.