A scaling characterization of nc-rank via unbounded gradient flow

Given a tuple of $n\times n$ complex matrices $\mathcal{A}=(A_1,A_2,\dots ,A_m)$, the linear symbolic matrix $A=A_1{x}_1+A_2{x}_2+\cdots +A_m{x}_m$ is nonsingular in the noncommutative sense if and only if the completely positive operators $T_{\mathcal{A}}(X)={\sum }_{i=1}^m{A}_{i}X{A}_i^{†}$ and $T_{\mathcal{A}}^{\ast }(X)={\sum }_{i=1}^m{A}_i^{†}X{A}_i$ can be scaled to be doubly stochastic: For every $\epsilon >0$ there are $g,h\in GL(n,\mathbb{C})$ such that $\mathrm{\parallel }{T}_{g^{†}\mathcal{A}h}(I)-I\mathrm{\parallel }<\epsilon$, $\mathrm{\parallel }{T}_{g^{†}\mathcal{A}h}^{\ast }(I)-I\mathrm{\parallel }<\epsilon$. In this paper, we show a refinement: The noncommutative corank of $A$ is equal to one-half of the minimum residual $\mathrm{\parallel }{T}_{g^{†}\mathcal{A}h}(I)-I{\mathrm{\parallel }}_1+\mathrm{\parallel }{T}_{g^{†}\mathcal{A}h}^{\ast }(I)-I{\mathrm{\parallel }}_1$ over all possible scalings $g^{†}\mathcal{A}h$, where $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_1$ is the trace norm. To show this, we interpret the residuals as gradients of a convex function on symmetric space $GL(n,\mathbb{C})\mathrm{/}{U}_n$, and establish a general duality relation of the minimum gradient-norm of a lower-unbounded convex function $f$ on $GL(n,\mathbb{C})\mathrm{/}{U}_n$ with an invariant Finsler metric, by utilizing the unbounded gradient flow of $f$ at infinity.