Paper 1, Section II, B

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

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

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

$3 x^{2}+2 x y+2 y^{2}+2 x z+2 z^{2}=1$

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

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

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

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

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

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