A scaling characterization of nc-rank via unbounded gradient flow

TL;DR

This paper characterizes the nc-rank of matrix tuples through a refined scaling approach using unbounded gradient flow, linking noncommutative rank to residuals of scaled completely positive operators.

Contribution

It introduces a new characterization of nc-rank via minimum residuals of scaled operators, interpreted through convex analysis and unbounded gradient flow on symmetric spaces.

Findings

Noncommutative corank equals half the minimal residual over scalings.

Residuals are interpreted as gradients of a convex function.

Establishes a duality relation using unbounded gradient flow.

Abstract

Given a tuple of $n \times n$ complex matrices ${\cal A} = (A_1,A_2,\ldots, A_m)$, the linear symbolic matrix $A = A_1x_1 + A_2x_2 + \cdots + A_m x_m$ is nonsingular in the noncommutative sense if and only if the completely positive operators $T_{\cal A} (X) = \sum_{i=1}^m A_i X A_i^{\dagger}$ and $T_{\cal A}^*(X) = \sum_{i=1}^m A_i^{\dagger} X A_i$ can be scaled to be doubly stochastic: For every $\epsilon > 0$ there are $g,h \in GL(n,\mathbb{C})$ such that $\|T_{g^{\dagger}{\cal A}h}(I)- I\| < \epsilon$, $\| T^*_{g^\dagger{\cal A}h}(I) - I\| < \epsilon$. In this paper, we show a refinement: The noncommutative corank of $A$ is equal to one-half of the minimum residual $\|T_{g^{\dagger}{\cal A}h}(I) - I\|_1 + \|T^*_{g^{\dagger}{\cal A}h}(I) - I\|_1$ over all possible scalings $g^{\dagger}{\cal A}h$, where $\|\cdot \|_1$ is the trace norm. To show this, we interpret the residuals as gradients of a convex function on symmetric space $GL(n,\mathbb{C})/U_n$, and establish a general duality relation of the minimum gradient-norm of a lower-unbounded convex function $f$ on $GL(n,\mathbb{C})/U_n$ with an invariant Finsler metric, by utilizing the unbounded gradient flow of $f$ at infinity.