MATH125 Lecture Notes - Lecture 36: Asteroid Family

• School: University of Alberta
• Department: Mathematics
• Course: MATH125
• Professor: Nikita

Document Summary

Show that the following matrix is orthogonal and nd its inverse: sin cos ] Cos sin + sin cos sin2 + cos2 . Bt b =[ cos2 + sin2 . Sin cos + cos sin . Matrix b is the matrix of a rotation through the angle in r2. Any rotation is a length-preserving transformation, also called isometry. The next theorem shows that every orthogonal matrix trans- formation is an isometry. In fact, orthogonal matrices are characterized by either one of these two properties. Let q be an n n matrix. The following statements are equivalent: q is orthogonal, ||qx|| = ||x|| for every x in rn, qx qy = x y for every x and y in rn. Note that x y = xt y. (a) (c). Then qt q = i, and we have. Qx qy = (qx)t qy = xt qt qy = xt iy = xt y = x y. (c) (b).