Purchase Solution

Rotation matrices

Not what you're looking for?

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

Solution Summary

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

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 ...

Intro to the Physics Waves

Some short-answer questions involving the basic vocabulary of string, sound, and water waves.

Introduction to Nanotechnology/Nanomaterials

This quiz is for any area of science. Test yourself to see what knowledge of nanotechnology you have. This content will also make you familiar with basic concepts of nanotechnology.

Classical Mechanics

This quiz is designed to test and improve your knowledge on Classical Mechanics.

The Moon

Test your knowledge of moon phases and movement.