Paper 1, Section II, B

Vectors and Matrices | Part IA, 2012

(a) (i) Find the eigenvalues and eigenvectors of the matrix

A=(311120102)A=\left(\begin{array}{lll} 3 & 1 & 1 \\ 1 & 2 & 0 \\ 1 & 0 & 2 \end{array}\right)

(ii) Show that the quadric Q\mathcal{Q} in R3\mathbb{R}^{3} defined by

3x2+2xy+2y2+2xz+2z2=13 x^{2}+2 x y+2 y^{2}+2 x z+2 z^{2}=1

is an ellipsoid. Find the matrix BB of a linear transformation of R3\mathbb{R}^{3} that will map Q\mathcal{Q} onto the unit sphere x2+y2+z2=1x^{2}+y^{2}+z^{2}=1.

(b) Let PP be a real orthogonal matrix. Prove that:

(i) as a mapping of vectors, PP preserves inner products;

(ii) if λ\lambda is an eigenvalue of PP then λ=1|\lambda|=1 and λ\lambda^{*} is also an eigenvalue of PP.

Now let QQ be a real orthogonal 3×33 \times 3 matrix having λ=1\lambda=1 as an eigenvalue of algebraic multiplicity 2. Give a geometrical description of the action of QQ on R3\mathbb{R}^{3}, giving a reason for your answer. [You may assume that orthogonal matrices are always diagonalisable.]

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