Past exam of the mathematical methods I course of the University of Oxford / 2019 / 11 / a / Solution

For a finite matrix, the series defining the matrix exponential converges absolutely in any matrix norm, because $\Vert A^m/m!\Vert \le \Vert A{\Vert }^m/m!$. Thus the Hermitian adjoint may be taken term by term. Since $A^{†}=A$,Therefore the matrix exponential of a Hermitian matrix is Hermitian.

If $A=\mathrm{diag}(a_1,\dots ,a_n)$ is a diagonal matrix, then $A^m=\mathrm{diag}(a_1^m,\dots ,a_n^m)$. Summing each entry separately givesFor the Hermitian matrix case each $a_i$ is real. The diagonal entry formula itself remains valid for complex diagonal entries.