I think you @Tamas_Papp should tell us a little more about your `Q`

matrix and how many different `t`

values you need.

Maybe you can just sample 5 different ones from your application and post them?

Also, what is the thing that `Q`

represents?

My mental model is that you have markov process transition probabilities in some graph of states, with possibly some structural constraints (e.g. sparsity, automorphisms, etc) and numerical values that are “generic” (i.e. not fine-tuned to sit at some special critical value) and not huge (i.e. `norm(Q)`

is not giant). My mental model is that the transition graph is strongly connected (otherwise you’d have split it up already!).

Generally, you know that you have a leading eigenvalue of `0`

(your stationary distribution), and the other ones are all stable (negative real part), and likely many will be quite stable (largish negative real parts).

Absent structural reasons you can expect your eigenspaces to be of dimension 1, with non-tiny spectral gaps (random matrix theory: eigenvalues repel) – so diagonalization would be possible without numerical blow-up.

If you conceptually come from discretized PDE, then your `Q`

is an approximation of an unbounded operator and you don’t want to apply it to vectors (you would need stiff solvers in ODE-speak).

If you need a lot of different `t`

values, then the tradeoffs change in terms of “how much time can we spend on matrix factorization”.