 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-1/5/ii/b/solution

The convergent power series defining the matrix exponential gives

e^{A} = U e^{D} U^{†}

, with

e^{D} = diag (e^{λ_{1}}, \dots, e^{λ_{n}})

det (e^{A}) = j = 1 \prod n e^{λ_{j}} = e^{\sum_{j} λ_{j}} = e^{tr A} .

The diagonalization proof uses the given Hermitian hypothesis; the identity itself holds for every square real or complex matrix.

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