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Unformatted text preview: Solution for Assignment #1 Khurram Hassan-Shafique To prove statements of Q.2, I have first proved that a matrix A is a rotation matrix if and only if it is orthonormal and det A = 1 (There may be other ways to prove the same statements). The definitions and other properties used in the proofs are given below: Definition: An n n matrix A over R is said to be orthonormal iff A t A = I . Equivalently, A- 1 = A t . Lemma: The following are equivalent. ( i ) A is orthonormal. ( ii ) Ax.Ay = x.y for all vectors x and y . ( iii ) The columns of A are mutually orthogonal unit vectors. Proof. The equivalence of (i) and (iii) is trivial. To prove that (i) implies (ii), assume that A is orthonormal then Ax.Ay = ( Ax ) t ( Ay ) = x t A t Ay = x t Iy = x t y = x.y To prove that (ii) implies (i), assume that Ax.Ay = x.y for all x and y , then we have e i .e j = e i .e j e t i e j = Ae i .Ae j e t i Ie j = e t i A t Ae j e t i Ie j- e t i A t Ae j = 0 e t i ( I- A t A ) e j = 0 I- A t A = 0 A t A = I Therefore A is orthonormal. Definition: Let SO n be the set of all n n orthonormal matrices A such that det A = 1. Theorem: A is a 2 dimensional rotation matrix if and only if A SO 2 . Proof: = Let A be a 2-D rotation matrix, then A t A = " cos - sin sin cos #" cos sin - sin cos # = " cos 2 + sin 2 cos sin - sin cos sin cos - cos sin sin 2 + cos 2 # = " 1 0 0 1 # = I 1...
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This note was uploaded on 06/12/2011 for the course CAP 5415 taught by Professor Staff during the Fall '08 term at University of Central Florida.
- Fall '08