Algebra and Geometry | Part IA, 2006

(i) Let u,v\mathbf{u}, \mathbf{v} be unit vectors in R3\mathbb{R}^{3}. Write the transformation on vectors xR3\mathbf{x} \in \mathbb{R}^{3}

x(ux)u+v×x\mathbf{x} \mapsto(\mathbf{u} \cdot \mathbf{x}) \mathbf{u}+\mathbf{v} \times \mathbf{x}

in matrix form as xAx\mathbf{x} \mapsto A \mathbf{x} for a matrix AA. Find the eigenvalues in the two cases (a) when uv=0\mathbf{u} \cdot \mathbf{v}=0, and (b) when u,v\mathbf{u}, \mathbf{v} are parallel.

(ii) Let M\mathcal{M} be the set of 2×22 \times 2 complex hermitian matrices with trace zero. Show that if AMA \in \mathcal{M} there is a unique vector xR3\mathrm{x} \in \mathbb{R}^{3} such that

A=R(x)=(x3x1ix2x1+ix2x3)A=\mathcal{R}(\mathbf{x})=\left(\begin{array}{cc} x_{3} & x_{1}-i x_{2} \\ x_{1}+i x_{2} & -x_{3} \end{array}\right)

Show that if UU is a 2×22 \times 2 unitary matrix, the transformation

AU1AUA \mapsto U^{-1} A U

maps M\mathcal{M} to M\mathcal{M}, and that if U1R(x)U=R(y)U^{-1} \mathcal{R}(\mathbf{x}) U=\mathcal{R}(\mathbf{y}), then x=y\|\mathbf{x}\|=\|\mathbf{y}\| where \|\cdot\| means ordinary Euclidean length. [Hint: Consider determinants.]

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