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Unformatted text preview: CALIFORNIA INSTITUTE OF TECHNOLOGY Department of Mathematics Math 1b; Solutions to Homework Set 7 Due: March 3, 2008 1. Let A = a b c d be a real 2 by 2 matrix. Then A is orthogonal iff AA t = I , or equivalently A t = A- 1 . Further as A is real, Theorem 5.19 says det( A ) = = 1. Now from Problem 4 on HW1, det( A ) = ad- bc and A- 1 = d- b- c a . Thus A is orthogonal iff a c b d = A t = A- 1 = d- b- c a , which holds iff a = d and b =- c . That is A is orthogonal iff A = a- c c a and a 2 + c 2 = 1 since = det( A ) = ( a 2 + c 2 ). As a and c are real, this last equation is equivalent to a = cos( ) and c = sin( ) for some angle . Therefore (a) is established, since = 1 if A is proper. Next A is improper iff =- 1 and hence A is improper iff A = cos( ) sin( ) sin( )- cos( ) . Thus we have our description of improper matrices. Further the two matrices in (b) correspond to the cases = 0 and 180 degrees, respectively, so they are improper. 2. (a) First char A ( x ) = x 2- tr ( A ) + det( A ) = x 2 + a 2 = ( x- ia )( x + ia ), so the eigenvalues of A are ia , and they are distinct as...
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