Krdytkyu

For any two symmetric positive definite (SPD) matrices $A$ and $B$, the Affine Invariant Riemannian Metric (AIRM) between them is defined as [1], [2]:

$d(A,B)=||log (A^{-1/2}B A^{-1/2}) ||_F$,

where $log(A)$ is the matrix logrithm of $A$, and $||A||_F$ is the Frobenius norm of $A$.

On the other hand, it is also shown that (e.g., see [3])

$d(A,B)=||log (A^{-1/2}B A^{-1/2}) ||_F = ||log (A^{-1}B) ||_F$.

However, when I used Matlab function logm, I found that $||log (A^{-1/2}B A^{-1/2}) ||_F ne ||log (A^{-1}B) ||_F$ at all (but their eigenvalues are the same).

I was wondering if there is anything wrong with my understanding the definition of AIRM?

Thanks very much!

[1] R. Bhatia, Positive Definite Matrices. Princeton University Press, 2009.

[2] X. Pennec, P. Fillard, and N. Ayache, “A Riemannian framework for tensor computing,” International Journal of Computer Vision, vol. 66, no. 1, pp. 41–66, 2006.

[3] M. Moakher, “A differential geometric approach to the geometric mean of symmetric Positive-Definite matrices,” SIAM J. Matrix Anal. Appl., vol. 26, no. 3, pp. 735–747, 2005.

[4] I. Horev, F. Yger, and M. Sugiyama, “Geometry-aware principal component analysis for symmetric positive definite matrices,” Machine Learning, 2017.