 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2023/digital-systems-linear-algebra-and-introduction-to-formal-proof/6/f/solution

λ

, choose a nonzero real eigenvector

v

with

A v = λ v

. By (e),

0 = v^{⊤} A v = λ v^{⊤} v .

Since

v^{⊤} v = ∥ v ∥^{2} > 0

, $λ = 0$ . If eigenvectors were initially allowed to be complex, the real matrix

A - λ I

has a nonzero real null space whenever it has a nonzero complex null space, so this choice of

v

remains justified.

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