On the zero set of the Kobayashi--Royden pseudometric of the spectral unit ball

TL;DR

This paper characterizes the zero set of the Kobayashi--Royden pseudometric on the spectral unit ball, linking tangent vectors to entire curves and providing a complete description for the case of 3x3 matrices.

Contribution

It establishes a precise criterion for tangent vectors at a point in the spectral unit ball and fully describes the zero set of the pseudometric in the 3x3 case.

Findings

Characterizes tangent vectors as elements of the tangent cone to the isospectral variety.

Provides a complete description of the zero set of the pseudometric for 3x3 matrices.

Links geometric properties of the spectral ball to complex analysis and matrix theory.

Abstract

Given $A\in\Omega_n,$ the $n^2$-dimensional spectral unit ball, we show that $B$ is a "generalized" tangent vector at $A$ to an entire curve in $\Omega_n$ if and only if $B$ is in the tangent cone $C_A$ to the isospectral variety at $A.$ In the case of $\Omega_3,$ the zero set of this metric is completely described.