Paper 1, Section II, B

Vectors and Matrices | Part IA, 2013

(a) Let AA and AA^{\prime} be the matrices of a linear map LL on C2\mathbb{C}^{2} relative to bases B\mathcal{B} and B\mathcal{B}^{\prime} respectively. In this question you may assume without proof that AA and AA^{\prime} are similar.

(i) State how the matrix AA of LL relative to the basis B={e1,e2}\mathcal{B}=\left\{\mathbf{e}_{1}, \mathbf{e}_{2}\right\} is constructed from LL and B\mathcal{B}. Also state how AA may be used to compute LvL \mathbf{v} for any vC2\mathbf{v} \in \mathbb{C}^{2}.

(ii) Show that AA and AA^{\prime} have the same characteristic equation.

(iii) Show that for any k0k \neq 0 the matrices

(acbd) and (ac/kbkd)\left(\begin{array}{ll} a & c \\ b & d \end{array}\right) \text { and }\left(\begin{array}{cc} a & c / k \\ b k & d \end{array}\right)

are similar. [Hint: if {e1,e2}\left\{\mathbf{e}_{1}, \mathbf{e}_{2}\right\} is a basis then so is {ke1,e2}\left\{k \mathbf{e}_{1}, \mathbf{e}_{2}\right\}.]

(b) Using the results of (a), or otherwise, prove that any 2×22 \times 2 complex matrix MM with equal eigenvalues is similar to one of

(a00a) and (a10a) with aC.\left(\begin{array}{ll} a & 0 \\ 0 & a \end{array}\right) \text { and }\left(\begin{array}{ll} a & 1 \\ 0 & a \end{array}\right) \text { with } a \in \mathbb{C} .

(c) Consider the matrix

B(r)=12(1+r1r11r1+r1112r)B(r)=\frac{1}{2}\left(\begin{array}{ccc} 1+r & 1-r & 1 \\ 1-r & 1+r & -1 \\ -1 & 1 & 2 r \end{array}\right)

Show that there is a real value r0>0r_{0}>0 such that B(r0)B\left(r_{0}\right) is an orthogonal matrix. Show that B(r0)B\left(r_{0}\right) is a rotation and find the axis and angle of the rotation.

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