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 Incoming links: Exponential of a skew-Hermitian matrix

A^{†} = - A

, then

H = - i A

H^{†} = i A^{†} = - i A = H

Q

with

H = Q diag (θ_{1}, \dots, θ_{n}) Q^{†}, θ_{j} \in R .

Thus

A = Q diag (i θ_{j}) Q^{†}

e^{A} = QC Q^{†}, C = diag (e^{i θ_{1}}, \dots, e^{i θ_{n}}) .

Each diagonal entry has modulus one, so

C^{†} C = I

. Using

Q^{†} Q = Q Q^{†} = I

(e^{A})^{†} e^{A} = Q C^{†} Q^{†} QC Q^{†} = I .

Hence the exponential of a skew-Hermitian matrix is unitary.