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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2022

Paper 1

\b[No.] Consider the two orthogonal projections
$$
P=\begin{pmatrix}1&0\\0&0\end{pmatrix},
\qquad
Q=\frac12\begin{pmatrix}1&1\\1&1\end{pmatrix}.
$$
Their product is
$$
PQ=\frac12\begin{pmatrix}1&1\\0&0\end{pmatrix}.
$$
For $x=(-1,2)^T$,
$$
x^\dagger PQx=-\frac12<0.
$$
\b[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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Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 1 / 7B / e / Solution

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

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

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