Vectors and Matrices | Part IA, 2008

Prove that the eigenvalues of a Hermitian matrix are real and that eigenvectors corresponding to distinct eigenvalues are orthogonal (i.e. eiej=0\mathbf{e}_{i}^{*} \cdot \mathbf{e}_{j}=0 ).

Let AA be a real 3×33 \times 3 non-zero antisymmetric matrix. Show that iAi A is Hermitian. Hence show that there exists a (complex) eigenvector e1\mathbf{e}_{1} such Ae1=λe1A \mathbf{e}_{1}=\lambda \mathbf{e}_{1}, where λ\lambda is imaginary.

Show further that there exist real vectors u\mathbf{u} and v\mathbf{v} and a real number θ\theta such that

Au=θv and Av=θuA \mathbf{u}=\theta \mathbf{v} \quad \text { and } \quad A \mathbf{v}=-\theta \mathbf{u}

Show also that AA has a real eigenvector e3\mathbf{e}_{3} such that Ae3=0A \mathbf{e}_{3}=0.

Let R=I+n=1Ann!R=I+\sum_{n=1}^{\infty} \frac{A^{n}}{n !}. By considering the action of RR on u,v\mathbf{u}, \mathbf{v} and e3\mathbf{e}_{3}, show that RR is a rotation matrix.

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