## Nielsen and Chuang Exercise 2.60

Exercise 2.60Show that has eigenvalues , and that the projectors on to the corresponding eigenspace are given by .

Starting from the spectral decomposition

and squaring both sides yields

(The second-to-last step uses the result from Exercise 2.35 that .) Since the eigenvalues of the Hermitian matrix are real, .

We can easily verify that meets the requirements for the project that , that , and that . Thus, is a set of projectors for .

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