Algebra and Geometry | Part IA, 2005

Let nR3\mathbf{n} \in \mathbb{R}^{3} be a unit vector. Consider the operation

xn×x\mathbf{x} \mapsto \mathbf{n} \times \mathbf{x}

Write this in matrix form, i.e., find a 3×33 \times 3 matrix A\mathbf{A} such that Ax=n×x\mathbf{A} \mathbf{x}=\mathbf{n} \times \mathbf{x} for all x\mathbf{x}, and compute the eigenvalues of A\mathbf{A}. In the case when n=(0,0,1)\mathbf{n}=(0,0,1), compute A2\mathbf{A}^{2} and its eigenvalues and eigenvectors.

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