 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/ia/paper-1/2b/solution

For a complex matrix, its Hermitian conjugate is obtained by transposing and taking entrywise complex conjugates:

(A^{†})_{ij} = \overline{A_{ji}}, A^{†} = \overline{A}^{T} .

The defining conditions are

A is Hermitian ⟺ A^{†} = A, A is unitary ⟺ A^{†} A = A A^{†} = I .

For a square matrix, either one of the two inverse identities in the unitary matrix condition implies the other. A Hermitian matrix may have complex entries, but its diagonal entries are real.

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