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Lecture 36

# MATH125 Lecture 36: 36

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School
University of Alberta
Department
Mathematics
Course
MATH125
Professor
Nikita
Semester
Winter

Description
Lecture 36, April 10, 2017. Example. Show that the following matrix is orthogonal and ▯nd its inverse: [ ] B = cos▯ ▯sin▯ sin▯ cos▯ Solution. [ 2 2 ] [ ] B B = cos ▯ + sin ▯ ▯cos▯ sin▯ + sin▯ cos▯ = 1 0 ▯sin▯ cos▯ + cos▯ sin▯ sin ▯ + cos ▯ 0 1 Therefore B is orthogonal and [ ] ▯1 T cos▯ sin▯ B = B = ▯sin▯ cos▯ Remark. Matrix B is the matrix of a rotation through the angle ▯ in R . Any rotation is a length-preserving transformation, also called isometry. The next theorem shows that every orthogonal matrix trans- formation is an isometry. Orthogonal matrices also preserve dot prod- ucts. In fact, orthogonal matrices are characterized by either one of these two properties. Theorem. Let Q be an n ▯ n matrix. The following statements are equivalent: a.Q is orthogonal. b. ▯▯Qx▯▯ = ▯▯x▯▯ for every x in R . n c. Qx ▯ Qy = x ▯ y for every x and y in R . T Proof. Note that x ▯ y = x y. (a) ▯ (c). Assume that Q is orthogonal. Then Q Q = I, and we have Qx ▯ Qy = (Qx) Qy = x Q Qy = x Iy = x y = x ▯ y: (c) ▯ (b). If Qx▯Qy = x▯y for every x;y, then, taking y = x, we have Qx ▯ Qx = x ▯ x, so √ ▯ ▯▯Qx▯▯ = Qx ▯ Qx = x ▯ x = ▯▯x▯▯: (b) ▯ (c). Assume that (b) holds and let q ienote the ith column of Q. We have 1 1 x ▯ y = (▯▯x + y▯▯ ▯ ▯▯x ▯ y▯▯ ) (▯▯Q(x +
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