  Normal matrices, or matrices which commute with their adjoints, are of
fundamental importance in pure and applied mathematics. In this paper, we study
a natural functional on the space of square complex matrices whose global
minimizers are normal matrices. We show that this functional, which we refer to
as the non-normal energy, has incredibly well-behaved gradient descent
dynamics: despite it being non-convex, we show that the only critical points of
the non-normal energy are the normal matrices, and that its gradient descent
trajectories fix matrix spectra and preserve the subset of real matrices. We
also show that, even when restricted to the subset of unit Frobenius norm
matrices, the gradient flow of the non-normal energy retains many of these
useful properties. This is applied to prove that low-dimensional homotopy
groups of spaces of unit norm normal matrices vanish; for example, we show that
the space of $d \times d$ complex unit norm normal matrices is simply connected
for all $d \geq 2$. Finally, we consider the related problem of balancing a
weighted directed graph -- that is, readjusting its edge weights so that the
weighted in-degree and out-degree is the same at each node. We adapt the
non-normal energy to define another natural functional whose global minima are
balanced graphs and show that gradient descent of this functional always
converges to a balanced graph, while preserving graph spectra and realness of
the weights. Our results were inspired by concepts from symplectic geometry and
Geometric Invariant Theory, but we mostly avoid invoking this machinery and our
proofs are generally self-contained.
