practiceprob4_soln

practiceprob4_soln - Math 235 Assignment 4 Practice...

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Math 235 Assignment 4 Practice Problems Solutions 1. Prove that the product of two orthogonal matrices is an orthogonal matrix. Solution: Let P and Q be orthogonal matrices. Then we have ( PQ ) T ( PQ ) = Q T P T PQ = Q T Q = I, since P T P = I and Q T Q = I . Thus PQ is also orthogonal. 2. (a) Recall that the dot product of two vectors ~x and ~ y can be written as ~x · ~ y = ± x 1 ··· x n ² y 1 . . . y n = ~x T ~ y use this fact to show that if an n × n matrix P is orthogonal, then k P~x k = k ~x k for every ~x R n . Solution: Suppose that P is orthogonal, then P T P = I . Then for any ~x R n we have k P~x k 2 = ( P~x ) · ( P~x ) = ( P~x ) T ( P~x ) = ( ~x T P T )( P~x ) = ~x T ( P T P ) ~x = ~x T ~x = k ~x k 2 . Hence k P~x k = k ~x k for every ~x . (b) Show that any real eigenvalue of an orthogonal matrix must be either 1 or - 1. Solution: Let λ be a real eigenvalue of P with corresponding eigenvector ~x . Then, using (a) we get k ~x k = k P~x k = k λ~x k = | λ
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practiceprob4_soln - Math 235 Assignment 4 Practice...

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