«Problem 3.17 The 2 x 2 matrix representing a rotation of the xy-plane is T=(cos

Question

«Problem 3.17 The 2 x 2 matrix representing a rotation of the xy-plane is T=(cos sin -sin cos Show that (except for certain special angles-what are they?) this matrix has no real eigenvalues. (This reflects the geometrical fact that no vector in the plane is carried into itself under such a rotation; contrast rotations in three dimensions.) This matrix does, however, have complex eigenvalues and eigenvectors. Find them. Construct a matrix S which diagonalizes T. Perform the similarity transformation (STS) explicitly, and show that it reduces T to diagonal form.

Explanation / Answer

sum of Eigen value say A+B= trace=cos theta+ cos theta

So A+B= 2 cos theta

Also product of Eigen value is equal to determinantsthus A*B=cos^2theta+ sin^2yheta=1

A*B=1

Or B= 1/A

Thus A+1/A= 2 cos theta

A^2+1= 2A cos theta

A^2-2A cos theta+1=0

The solution of this equation is not to be real but complex and thus the Eigen value of the given matrices are complex

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