I agree with your criticism. The point I awkwardly tried to make is this. Computing a matrix exponential and computing the singular part of the quotient of the exponential function by a nonzero polynomial are equivalent problems (in the sense that there is a simple dictionary between them).

I looked at Chapter 10, but I still don't see your point. I wrote "Let f(z) be the minimal polynomial of A, let g(z) be f(z) times the singular part of e^z/f(z), and observe e^A=g(A)." Are you claiming that Higham disproves this statement? In this case, could you say where? Are you claiming that this statement is false? In this case, could you say why?

Dear Robin, Thanks for your answer. I don't have Nick Higham's book. I was wondering if you could be more precise. Your comment is very surprising to me: I thought I was stating a triviality. Here are two references en.wikipedia.org/wiki/Matrix_exponential#Alternative iecn.u-nancy.fr/~gaillard/DIVERS/Constant_coefficients Looking forward to hearing from you.

Dear Johannes, please reread my post.

Thank you! Unfortunately, I don't understand Adrian Mathias' review. But I feel I understand Bourbaki's book.