 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2002/paper-1/6/b/solution

For

u = (1, 1)^{T}

and

v = (1, - 1)^{T}

u u^{T} = (1111), v v^{T} = (1 - 1 - 1 1) .

Their inner product is zero and both squared norms are two. Thus

P_{u} = u u^{T} /2

and

P_{v} = v v^{T} /2

are orthogonal spectral projectors, with

P_{u} + P_{v} = I

. Since

A u = 5 u

and

A v = v

λ = 5

μ = 1

, and

A = \frac{5}{2} u u^{T} + \frac{1}{2} v v^{T}, A^{- 1} = \frac{1}{10} u u^{T} + \frac{1}{2} v v^{T} .

Multiplying these decompositions uses

P_{u} P_{v} = 0

and

P_{u}^{2} = P_{u}

P_{v}^{2} = P_{v}

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