Science talk:Math Exam Resources/Courses/MATH152/April 2015/Question B 6 (a)

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Use geometry119:22, 17 March 2018

Use geometry

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Last edit: 19:22, 17 March 2018

ZIMINGYIN:

The solution is correct, but excessively complicated and ignores the geometry of the question.

One should first determine the angle of rotation about the axis. The eigenvalues are then given by (Note that this implies that the trace of the matrix is )

Nicholas Hu (talk)06:10, 17 March 2018

But an even more elegant solution for this particular problem is as follows (sketched below; please fill in the details):

A rotation matrix is orthogonal, and therefore has determinant ; it is easy to see in this case that the determinant is 1. One of the eigenvalues must be 1 (it's a rotation in an odd-dimensional space); let the two others be . Since the sum of the eigenvalues is the trace of the matrix, and since their product is the determinant, we have and , whence

Nicholas Hu (talk)06:36, 17 March 2018