 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2016-resit/9/ii/solution

The

y

coordinate is an independent eigen-direction, while the

x z

block is

(2112)

. Thus

det (A - λ I) = (1 - λ)^{2} (3 - λ),

which gives

det (A - I) = 0

λ = 1 : span {(1, 0, - 1), (0, 1, 0)}; λ = 3 : span {(1, 0, 1)} .

Divide the two diagonal

x z

2

to obtain an orthonormal eigenbasis. The eigenvalue one has multiplicity two; giving only the

y

axis would omit an independent eigenvector.

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