An early-ish paper$^1$ on the Riemannian structure of the manifold $\mathcal{P}(n)$ of symmetric, positive-definite matrices gives the following formula for the geodesic distance between two such matrices:
$$ \begin{align*} d^2_{\mathcal{P}(n)}(P_1,P_2) = \Vert \log(P_1^{-1}P_2) \rVert^2_F = \sum_{i=1}^n \ln^2(\lambda_i), &&&&&&(2.9) \end{align*} $$
where "$\log$" is here understood as the matrix logarithm and $\lambda_i$ are the eigenvalues of $P^{-1}P_2$. However, the pyRiemann implementation of this distance (which cites this paper) documents the computation of this distance as
$$ d^2_{\mathcal{P}(n)}(P_1,P_2) = \sum_{i=1}^n \ln^2(\lambda_i), $$
where $\lambda_i$ are the joint eigenvalues of $P_1$ and $P_2$. With a cursory search, I did not find a result showing that the joint eigenvalues of two matrices $A$ and $B$ are the same as the eigenvalues of $A^{-1} B$, and it is not immediately obvious to me if that's true or why it would be. Are these two spectra actually the same (or is there some straightforward relationship between them?), or is there an inaccuracy in the implementation? Thanks!
- Moakher, Maher, A differential geometric approach to the geometric mean of symmetric positive-definite matrices, SIAM J. Matrix Anal. Appl. 26, No. 3, 735-747 (2005). ZBL1079.47021.