Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 1 / 18 / b

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- Solution b

Solution

For a matrix with equal diagonal entries, the symmetric and antisymmetric directions

(1, 1)

and

(1, - 1)

are eigenvectors. Direct multiplication of

A A^{T}

gives eigenvalues

3/2 - 1/2 = 1

and

3/2 + 1/2 = 2

. Dividing each direction by its Euclidean norm gives

λ_{1} = 1, u_{1} = \frac{1}{2} (11); λ_{2} = 2, u_{2} = \frac{1}{2} (1 - 1) .

The two normalized eigenvectors are orthogonal; either may independently be replaced by its negative.

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