Past exam of the year 1 of the computer science course of the University of Oxford / 2013 / Digital systems, linear algebra, introduction to formal proof / 6 / d

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- Solution d

Solution

Use the characteristic polynomial convention

p (λ) = det (λ I - P)

, with

I

the identity matrix. Put

u = λ - 1/2

. Then

p (λ) = det u - 1/4 - 1/4 - 1/2 u 0 - 1/2 0 u = u^{3} - \frac{1}{8} u - \frac{1}{8} u = u (u^{2} - \frac{1}{4}) = λ (λ - 1/2) (λ - 1) .

The required characteristic polynomial equation for this matrix is therefore

p (λ) = λ^{3} - \frac{3}{2} λ^{2} + \frac{1}{2} λ = 0.

Using

det (P - λ I)

instead negates the cubic polynomial and leaves the eigenvalues unchanged.

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