# Is the matrix exponential independent of the norm?

The definition of the matrix exponential uses powers, multiplication by scalars, sums and a limit. The former 3 are given by the vector space the matrices are in. A limit would require making that space of matrices into a normed space, i believe. Since there are different possible norms for a space of matrices, it is not clear that the same sequences converge for each norm, which could mean that the matrix exponential depends on norm.

While writing this i remembered that a space of matrices is isomorphic (as a vector space) to $$\mathbb{F}^k$$ for some $$k \in \mathbb{N}$$ and some field $$\mathbb{F}$$ and all norms in $$\mathbb{F}^k$$ are equivalent. I believe isomorphism as a vector space would imply that all norms are equivalent since norms are defined in terms of the vector space. Is this correct?

(Edit: My last claim might be wrong if what someone commented is true, but its at least true for the real and complex fields)

• Yes. Convergence the series for exponential does not depend on the norm. Commented Sep 30, 2020 at 8:49
• Yes, in finite-dimensional spaces, all norms are equivalent.
– user436658
Commented Sep 30, 2020 at 9:00
• All norms on $\mathbb F^k$ are equivalent in the case that $\mathbb F$ is complete. So $\mathbb R$ and $\mathbb C$ are OK. It is an interesting exercise to find an example two norms on $\mathbb Q^2$ that are not equivalent. Commented Oct 1, 2020 at 0:07