Normal Matrices

$\operatorname{E}$ is not quasiconvex!

Theorem [with Needham]

The only critical points of $\operatorname{E}$ are the global minima; i.e., the normal matrices.

Gradient Descent

Let $\mathcal{F}: \mathbb{C}^{d \times d} \times \mathbb{R} \to \mathbb{C}^{d \times d}$ be negative gradient descent of $\operatorname{E}$; i.e.,

$\mathcal{F}(A_0,0) = A_0 \qquad \frac{d}{dt}\mathcal{F}(A_0,t) = -\nabla \operatorname{E}(\mathcal{F}(A_0,t))$.

Theorem [with Needham]

For any $A_0 \in \mathbb{C}^{d \times d}$, the matrix $A_\infty := \lim_{t \to \infty} \mathcal{F}(A_0,t)$ exists, is normal, has the same eigenvalues as $A_0$, and is real if $A_0$ is.

Why?

$\mathbb{C}^{d \times d}$ is symplectic, with symplectic form $\omega_A(X,Y) = -\mathrm{Im}\langle X,Y \rangle = -\mathrm{Im}\mathrm{Tr}(Y^\ast X)$.

Consider the conjugation action of $\operatorname{SU}(d)$ on $\mathbb{C}^{d \times d}$: $U \cdot A  = U A U^\ast$.

Geometric Invariant Theory (GIT)

The GIT quotient consists of group orbits which can be distinguished by $G$-invariant (homogeneous) polynomials.

Balancing Graphs

Define the unbalanced energy $\operatorname{B}(A) := \|\mathrm{diag}([A,A^\ast])\|^2 = \sum \left(\|A_i\|^2 - \|A^i\|^2\right)^2$.

Let $\mathscr{F}(A_0,0) = A_0, \frac{d}{dt}\mathscr{F}(A_0,t) = - \nabla \operatorname{B}(\mathscr{F}(A_0,t))$ be negative gradient flow of $\operatorname{B}$.

Balancing Graphs

Theorem (with Needham)

For any $A_0 \in \mathbb{C}^{d \times d}$, the matrix $A_\infty := \lim_{t \to \infty} \mathscr{F}(A_0,t)$ exists, is balanced, has the same eigenvalues and principal minors as $A_0$, and has zero entries whenever $A_0$ does.

If $A_0$ is real, so is $A_\infty$, and if $A_0$ has all non-negative entries, then so does $A_\infty$.

Balancing Graphs

Theorem (with Needham)

For any $A_0 \in \mathbb{C}^{d \times d}$, the matrix $A_\infty := \lim_{t \to \infty} \mathscr{F}(A_0,t)$ exists, is balanced, has the same eigenvalues and principal minors as $A_0$, and has zero entries whenever $A_0$ does.

If $A_0$ is real, so is $A_\infty$, and if $A_0$ has all non-negative entries, then so does $A_\infty$.

Balancing Graphs

Theorem (with Needham)

For any $A_0 \in \mathbb{C}^{d \times d}$, the matrix $A_\infty := \lim_{t \to \infty} \mathscr{F}(A_0,t)$ exists, is balanced, has the same eigenvalues and principal minors as $A_0$, and has zero entries whenever $A_0$ does.

If $A_0$ is real, so is $A_\infty$, and if $A_0$ has all non-negative entries, then so does $A_\infty$.

$\|A\|^2=1$

$\|A\|^2=0.569$

Balancing Graphs

Theorem (with Needham)

For any $A_0 \in \mathbb{C}^{d \times d}$, the matrix $A_\infty := \lim_{t \to \infty} \mathscr{F}(A_0,t)$ exists, is balanced, has the same eigenvalues and principal minors as $A_0$, and has zero entries whenever $A_0$ does.

If $A_0$ is real, so is $A_\infty$, and if $A_0$ has all non-negative entries, then so does $A_\infty$.

Balancing Graphs

Theorem (with Needham)

For any $A_0 \in \mathbb{C}^{d \times d}$, the matrix $A_\infty := \lim_{t \to \infty} \mathscr{F}(A_0,t)$ exists, is balanced, has the same eigenvalues and principal minors as $A_0$, and has zero entries whenever $A_0$ does.

If $A_0$ is real, so is $A_\infty$, and if $A_0$ has all non-negative entries, then so does $A_\infty$.

Preserving Weights

By doing gradient flow $\overline{\mathscr{F}}$ on the unit sphere, we can preserve weights:

Theorem (with Needham)

For any non-nilpotent $A_0 \in \mathbb{C}^{d \times d}$ with $\|A\|^2=1$, the matrix $A_\infty := \lim_{t \to \infty} \overline{\mathscr{F}}(A_0,t)$ exists, is balanced, has Frobenius norm 1, and has zero entries whenever $A_0$ does.

If $A_0$ is real, so is $A_\infty$, and if $A_0$ has all non-negative entries, then so does $A_\infty$.

Preserving Weights

By doing gradient flow $\overline{\mathscr{F}}$ on the unit sphere, we can preserve weights:

Theorem (with Needham)

For any non-nilpotent $A_0 \in \mathbb{C}^{d \times d}$ with $\|A\|^2=1$, the matrix $A_\infty := \lim_{t \to \infty} \overline{\mathscr{F}}(A_0,t)$ exists, is balanced, has Frobenius norm 1, and has zero entries whenever $A_0$ does.

If $A_0$ is real, so is $A_\infty$, and if $A_0$ has all non-negative entries, then so does $A_\infty$.

Preserving Weights

By doing gradient flow $\overline{\mathscr{F}}$ on the unit sphere, we can preserve weights:

Theorem (with Needham)

For any non-nilpotent $A_0 \in \mathbb{C}^{d \times d}$ with $\|A\|^2=1$, the matrix $A_\infty := \lim_{t \to \infty} \overline{\mathscr{F}}(A_0,t)$ exists, is balanced, has Frobenius norm 1, and has zero entries whenever $A_0$ does.

If $A_0$ is real, so is $A_\infty$, and if $A_0$ has all non-negative entries, then so does $A_\infty$.

Preserving Weights

By doing gradient flow $\overline{\mathscr{F}}$ on the unit sphere, we can preserve weights:

Theorem (with Needham)

For any non-nilpotent $A_0 \in \mathbb{C}^{d \times d}$ with $\|A\|^2=1$, the matrix $A_\infty := \lim_{t \to \infty} \overline{\mathscr{F}}(A_0,t)$ exists, is balanced, has Frobenius norm 1, and has zero entries whenever $A_0$ does.

If $A_0$ is real, so is $A_\infty$, and if $A_0$ has all non-negative entries, then so does $A_\infty$.