matrix exponential vs. matrix power, differentiability/stability/efficiency

[justktln2]

I'm carrying out a vectorized calculation of the connected components of a graph from its adjacency matrix. Now, the matrix power $A^n$ records the number of paths between nodes, and to check connectivity it is sufficient to check $A^{n-1}$ (where n is the dimension of the adjacency matrix), but in the end I'm going to be applying some kind of thresholding function (soft or hard), so I could also do this by thresholding the matrix exponential, $\exp(A)$. The matrices I'm working with will never be larger than 100 by 100 and will typically be sparse.

Computing the matrix power seems like it would be more efficient, but I wonder if there are any practical reasons to compute the matrix exponential instead (e.g. if jax.scipy.linalg.expm and/or its gradient is more efficient or better-behaved than jax.numpy.linalg.expm for algorithmic reasons that wouldn't be obvious to me as a biochemist.

[mygn-l]

Given a matrix A, we define matrix power to be A^n, while matrix exponentiation is defined as e^A, where e can be any base. They are not the same operation. You are probably looking for matrix power in your case. (Maybe I'm missing some deeper graph-theoretic insight between power and exponentiation though.)