Minimal curves in $Gl^{+}(n)$ with respect to the trace norms

GHIGLIONI, EDUARDO MARIO; ANTEZANA, JORGE ABEL; STOJANOFF, DEMETRIO

Encuentro; Mathematical Encounter; 2019

Consider the cone of $n imes n$ positive invertible matrices $Gl^{+}(n)$endowed with the bi-invariant Finsler metric given by the trace norm,$$|X|_{1,A}=|A^{-1/2}XA^{-1/2}|_1$$for any $X$ tangent to $Ain Gl^{+}(n)$. In this context, given two points $A, B in Gl^{+}(n)$, there exists infinitely many curves of minimal length. In this talk we will provide a characterization of such a curves. We get the characterization by lifting the problem to the space of hermitian matrices $H(n)$. So, firstly we provide a characterization of the minimal paths joining $X,Yin H(n)$, if the length of a curve $alpha:[a,b]o H(n)$ is measure by$$L(alpha)=int_a^b |dot{alpha}(t)|_1,dt.$$Once the characterization of the minimal curves is given for $H(n)$, the characterization in $Gl^{+}(n)$ can be obtained using the Exponential Metric Increasing property.