I am new to differential geometry and I am interested in finding an expression for the geodesic, $\gamma: [0,1] \to GL(n,\mathcal{C})$, connecting two complex matrices $A,B \in GL(n,\mathcal{C})$ as a function of $A$ and $B$.
So far I have approached it as follows:
If we set $\gamma(0) = A$ and $\dot{\gamma(0)} = X$, then this dissertation ,(https://etd.auburn.edu/bitstream/handle/10415/7093/Dissertation_ElaineGan.pdf?sequence=2&isAllowed=y), provides a nice closed-form expression for $\gamma$ as:
$\gamma_{A,X}(t) = Ae^{t(A^{-1}X)^*}e^{t(A^{-1}X - (A^{-1}X)^*)}$.
If we set
$\gamma_{A,X}(1) = Ae^{(A^{-1}X)^*}e^{A^{-1}X - (A^{-1}X)^*} = B$
and pre-multiply both sides by $A^{-1}$ to get
$e^{(A^{-1}X)^*}e^{A^{-1}X - (A^{-1}X)^*} = A^{-1}B $,
we could then try to solve for the initial velocity $X$ as a function of $A$ and $B$. However, I'm not sure how to proceed from here because I don't think we can apply the product formula for the matrix logarithm because I don't think $e^{(A^{-1}X)^*}$ and $e^{A^{-1}X - (A^{-1}X)^*}$ commute in general.
Does anyone know if there is a nice closed-form expression for this or if not, how I could proceed using the approach outlined above or a different approach?