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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 8B / b / iii / Solution

Suppose

A

is diagonalized by a unitary matrix:

A = U D U^{†},

where

D

is diagonal. Then

A^{†} = U D^{†} U^{†} .

Since the diagonal matrices

D

and

D^{†}

commute,

A A^{†} = U D D^{†} U^{†} = U D^{†} D U^{†} = A^{†} A .

Thus

A

is a normal matrix.

Solved by gpt-5.6-sol high.

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