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cos O - sin 0 Rot(0) = sin 0 cos O 1. Starting with the characteristic equation, solve for the two eigenvalues of Rot(0). 2. Find the corresponding eigenvectors of Rot(0) assuming that 0 is not a multiple of . Do the eigenvectors depend on 0? 3. For what values of 0 are the eigenvalues purely real? Explain how the rotation ends up "scaling" an input vector for these 0.

cos O - sin 0 Rot(0) = sin 0 cos O 1. Starting with the characteristic equation, solve for the two eigenvalues of Rot(0). 2. Find the corresponding eigenvectors of Rot(0) assuming that 0 is not a multiple of . Do the eigenvectors depend on 0? 3. For what values of 0 are the eigenvalues purely real? Explain how the rotation ends up "scaling" an input vector for these 0.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider once more the rotation matrix in two dimensions: Cos O - sin 0 - Rot(0) = sin 0 cos 0 1. Starting with the characteristic equation, solve for the two eigenvalues of Rot(0). 2. Find the corresponding eigenvectors of Rot(0) assuming that 0 is not a multiple of t. Do the eigenvectors depend on ? 3. For what values of 0 are the eigenvalues purely real? Explain how the rotation ends up "scaling" an input vector for these 0. 4. You should see that many "new" eigenvectors appear when the eigenvalues are real. Explain why these did not show up in part 2 even though you found that your solutions there did not depend on 0.
Transcribed Image Text:Consider once more the rotation matrix in two dimensions: Cos O - sin 0 - Rot(0) = sin 0 cos 0 1. Starting with the characteristic equation, solve for the two eigenvalues of Rot(0). 2. Find the corresponding eigenvectors of Rot(0) assuming that 0 is not a multiple of t. Do the eigenvectors depend on ? 3. For what values of 0 are the eigenvalues purely real? Explain how the rotation ends up "scaling" an input vector for these 0. 4. You should see that many "new" eigenvectors appear when the eigenvalues are real. Explain why these did not show up in part 2 even though you found that your solutions there did not depend on 0.
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