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Unformatted text preview: is a unit vector and D : The vector can be written as cos ./ sin ./ for some angle , so the orthogonal matrix has the form cos ./ sin ./ sin ./ cos ./ or cos ./ sin ./ sin ./ cos ./ . The matrix R D cos ./ sin ./ sin ./ cos ./ is the matrix of the rotation through an angle , and has determinant equal to 1 . The matrix cos ./ sin ./ sin ./ cos ./ equals R J , where J D 1 1 is the reection matrix J D J O e 2 . The determinant of R J is equal to 1 . Now consider the situation in three dimensions. Any real 3by3 matrix has a real eigenvalue, since the characteristic polynomial is cubic with real coefcients. A real eigenvalue of an orthogonal matrix must be 1 because the matrix implements an isometry. Lemma 4.4.12. Any element of SO . 3 ; R / has C 1 as an eigenvalue....
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 Fall '08
 EVERAGE
 Algebra, Determinant, Matrices

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