practiceprob4_soln

# practiceprob4_soln - Math 235 Assignment 4 Practice...

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

practiceprob4_soln - Math 235 Assignment 4 Practice...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online