Paper 1, Section II, B
(a) Show that a square matrix is anti-symmetric if and only if for every vector .
(b) Let be a real anti-symmetric matrix. Show that the eigenvalues of are imaginary or zero, and that the eigenvectors corresponding to distinct eigenvalues are orthogonal (in the sense that , where the dagger denotes the hermitian conjugate).
(c) Let be a non-zero real anti-symmetric matrix. Show that there is a real non-zero vector a such that .
Now let be a real vector orthogonal to . Show that for some real number .
The matrix is defined by the exponential series Express and in terms of and .
[You are not required to consider issues of convergence.]
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