= Exponential of a skew-Hermitian matrix
{title2=$A^\dagger=-A\implies(e^A)^\dagger e^A=I$}
If $A$ is <Skew-Hermitian>, then $-iA$ is a <Hermitian matrix>. The <finite-dimensional spectral theorem> gives $A=Q\operatorname{diag}(i\theta_j)Q^\dagger$, with real $\theta_j$ and a <unitary matrix> $Q$. Since the <matrix exponential respects similarity>, $e^A=Q\operatorname{diag}(e^{i\theta_j})Q^\dagger$. Each diagonal entry has modulus one, so $e^A$ is a <unitary matrix>. Thus a <skew-Hermitian matrix> generates norm-preserving linear evolution.
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