 Solution

ID: past-exam-of-the-numerical-linear-algebra-and-continuous-optimisation-course-of-the-university-of-oxford/2010/a/3/d/solution

For a unit eigenvector of

A

(A - I) x = (λ - 1) x

. The matrix 2-norm bound from part (a) gives

∣ λ - 1∣ \leq 2/3

. Hence every eigenvalue lies in

[α, β] = [1/3, 5/3] .

In particular the condition number in the Euclidean operator norm satisfies

κ_{2} (A) = λ_{m a x} / λ_{m i n} \leq 5

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