 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2022/digital-systems-linear-algebra-introduction-to-formal-proof-resit/5/d/solution

Let

A v = λ v

for a real eigenvalue

λ

and a nonzero real eigenvector

v

. Part (b) gives

λ \neq = 0

and

A^{- 1} v = λ^{- 1} v

. The expression for

B

in part (c) therefore gives

B v = (A + I + A^{- 1}) v = (λ + 1 + λ^{- 1}) v .

Consequently $λ + 1 + λ^{- 1}$ is a real eigenvalue of $B$ , with the same eigenvector

v

. For a real matrix and real eigenvalue, a nonzero real vector can be chosen in the real kernel of

A - λ I

; no complex-vector assumption is needed.

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