# When do two matrices have the same exponential?

Let $$A$$ and $$B$$ be $$n\times n$$ hermitean matrices. When do we have $$e^{iA}=e^{iB}$$? Can we somehow classify those pairs of matrices that have the same exponential?

Here are some observations that I made:

• If $$A$$ and $$B$$ commute, then the condition is satisfied if and only if the spectrum of $$A-B$$ is contained in $$2\pi\mathbb Z$$. (Both directions can fail if $$A$$ and $$B$$ don't commute, as the following two points show.)
• The condition $$e^{iA}=e^{iB}$$ can also be satisfied if $$A$$ and $$B$$ do not commute. Take, for example, $$A=2\pi\begin{pmatrix}1&0\\0&-1\end{pmatrix}$$ and $$B=2\pi\begin{pmatrix}0&1\\1&0\end{pmatrix}$$. They do not commute but $$e^{iA}=e^{iB}=I$$. The spectrum of their difference is not in $$2\pi\mathbb Z$$.
• If $$A=\sqrt2\pi\begin{pmatrix}1&0\\0&-1\end{pmatrix}$$ and $$B=\sqrt2\pi\begin{pmatrix}0&1\\1&0\end{pmatrix}$$, then the eigenvalues of $$A-B$$ are $$\pm2\pi$$ but $$e^{iA}\neq e^{iB}$$.
• The exponential map is not a homomorphism so finding the kernel is not enough; cf. the Baker–Campbell–Hausdorff formula.
• Having the same exponential is an equivalence relation.

Here is a rather cheap answer to this question. Let $E_\lambda$ denote the eigenspace of $\lambda \in \mathrm{spec}(A)$. Let $F_\lambda$ denote the eigenspace of $\lambda \in \mathrm{spec}(B)$. Then $\exp(iA) =\exp(iB)$ if and only if, for every $\lambda_0 \in \mathbb{R}$, we have $$\bigoplus_{\lambda \in \mathrm{spec}(A) \cap (\lambda_0 + 2 \pi \mathbb{Z})} E_\lambda = \bigoplus_{\lambda \in \mathrm{spec}(B) \cap (\lambda_0 + 2 \pi \mathbb{Z})} F_\lambda.$$
Roughly speaking, the condition is that $A$ and $B$ should have the same eigenspaces, after identifying the eigenvalues whose exponentials are equal.
• No problem! Of course, this isn't a particularly deep characterization. I guess a similar thing works if $A$ and $B$ are just normal and $f : \mathbb{C} \to \mathbb{C}$ is any continuous function (instead of $f(t) = e^{it}$). – Mike F May 1 '15 at 16:27