Let us consider a matrix manifold: $$ \mathcal{M} = \{ X \in \mathbb{R}^{M\times N} : C_1X = XC_2 \}, $$ where $C_1$ and $C_2$ are known (fixed) appropriately-dimensioned matrices. We can also endow the manifold with a classical Frobenius or trace metric, e.g. $<X,Y>=\mathrm{tr}(X^\top Y)$.
I would like to derive the operators to be used in Riemannian optimization, such as:
- The tangent space
- Projection onto the tangent space
- Exponential map
- Bonus: Logarithm maps and parallel transport.
Hoping that my question is clear enough, any thoughts on how to go about this?