Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 1 / 7B / e / Solution

No. Consider the two orthogonal projectionsTheir product isFor $x=(-1,2{)}^T$,Thus $PQ$ is not a positive semidefinite matrix. It is also not Hermitian; products of orthogonal projections need not remain orthogonal projections unless the factors commute.

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