Exponential representations of unitary matrices

We shall now demonstrate that any unitary matrix U can be written as the matrix exponential of an anti-Hermitian matrix  

We first show that, for any unitary matrix U, we can always find an anti-Hermitian matrix X such that (3.1.9) is satisfied. For this purpose, we recall that the spectral theorem states that any unitary matrix can be diagonalized as 

Since iV8V is anti-Hermitian, we have shown that any unitary matrix can be written in the exponential form (3.1.9). 

We have now satisfied the first of the three requirements for the unitary parametrization stated in the introduction to Section 3.1. To satisfy the second requirement, we note that, for any anti-Hermitian matrix X, the exponential exp(X) is always unitary since, from the relation X = —X, it follows that 

Finally, we note that the third requirement for the parametrization is also satisfied since anti-Hermitian matrices are trivially represented by a set of independent parameters. We may, for instance, take the matrix elements at the diagonal and below the diagonal as the independent ones and generate the remaining elements of the matrix from the anti-Hermitian condition = —Xgp. Note that the diagonal elements of an anti-Hermitian matrix are pure imaginary and that the off-diagonal elements are complex. 

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