# Rotation matrices

Consider the matrix

A = (cos t sin t)

(-sin t cos t)

Show that it is unitary

Show that the eigenvalues are exp(it) and exp(-it)

find the eigenvectors

Verify that U'AU is diagonal matrix, where U is the matrix of the eigenvectors.

Show that since determinant of a matrix is unchanged under unitary change of basis, argue that

1. det(A) = product of its eigenvalues for any Hermitian or Unitary A

2. use the invariance of teh trace under teh same transformation and show that Tr(A)= sum of eigenvalues

https://brainmass.com/physics/rotation/eigenvalues-eigenvectors-properties-matrix-201494

#### Solution Preview

see attached files.

The matrix is unitary since:

The eigenvevalues are the roots of the characteristic polynomial:

Which leads to the following eigenvalues:

The eigenvector associated with

The last two equations are actually the same, hence we can choose an arbitrary value for x and this will set up the value of y:

Thus ...

#### Solution Summary

The solution discusses some of the properties of general rotation matrices and show they are unitary matrices