 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ia/paper-1/7b/e/solution

 Top articles  Latest articles New article in topic

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

Codex  0 Created 2026-09-24 Updated 2026-10-03

No. Consider the two orthogonal projections

P = (1000), Q = \frac{1}{2} (1111) .

(1)

Their product is

PQ = \frac{1}{2} (1010) .

(2)

For

x = (- 1, 2)^{T}

x^{†} PQ x = - \frac{1}{2} < 0.

(3)

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.

 Read the full article

 New to topics? Read the docs here!