Unformatted text preview: ˇ ² is a unit vector and ± ± ı ² D ˙ ± ± ˇ ˛ ² : The vector ± ˛ ˇ ² can be written as ± cos .²/ sin .²/ ² for some angle ² , so the orthogonal matrix has the form ± cos .²/ ± sin .²/ sin .²/ cos .²/ ² or ± cos .²/ sin .²/ sin .²/ ± cos .²/ ² . The matrix R ± D ± cos .²/ ± sin .²/ sin .²/ cos .²/ ² is the matrix of the rotation through an angle ² , and has determinant equal to 1 . The matrix ± cos .²/ sin .²/ sin .²/ ± cos .²/ ² equals R ± J , where J D ± 1 ± 1 ² is the reﬂection 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 3–by–3 matrix has a real eigenvalue, since the characteristic polynomial is cubic with real coefﬁcients. 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....
View
Full Document
 Fall '08
 EVERAGE
 Linear Algebra, Algebra, Determinant, Matrices, Orthogonal matrix, special orthogonal group, det W

Click to edit the document details