Videos for Section 7.4


Rotations in $\mathbb{R}^n$ have some nice properties. If $R$ is a rotation matrix, then

  1. $R$ preserves length: $|R {\bf x}| = |{\bf x}|$.
  2. $R$ preserves inner products: $\langle R {\bf x} | R {\bf y} \rangle = \langle {\bf x} | {\bf y} \rangle$.
  3. The columns of $R$ are orthonormal.
  4. $R^T R = R R^T = I$. In other words, $R^T = R^{-1}$.
  5. The columns of $R^T$ (that is, the transposes of the rows of $R$) are orthonormal.
In fact, any matrix that satisfies one of these properties satisfies all of these properties. We call such matrices orthogonal. Orthogonal matrices have determinant $\pm 1$. The ones with determinant $+1$ are rotations, and the ones with determinant $-1$ are rotations followed by a reflection.

In the first video, we go through these properties, see why they are equivalent, and consider the complex analog.

An operator or matrix $U$ is unitary if $U^{-1}=U^\dagger$. Orthogonal matrices are unitary matrices that happen to be real. Unitary operators have properties that are the obvious extension of the properties of orthogonal matrices, namely (a) they preserve length, (b) they preserve inner products, (c) the columns of a unitary matrix are orthonormal, and so are the transposes of the rows.

The second video is about diagonalizing unitary operators or matrices (including orthogonal matrices). If $U$ is a unitary matrix (or operator on a finite-dimensional space), then

  1. All of the eigenvalues of $U$ lie on the unit circle: $|\lambda|=1$. It's often useful to write the eigenvalues as $\lambda_j = e^{i \theta_j}$.
  2. Eigenvectors with different eigenvalues are orthogonal.
  3. $U$ is diagonalizable.
  4. You can find an orthonormal basis for the vector space consisting of eigenvectors of $U$.
These properties should remind you of diagonalizing Hermitian operators (or matrices). The only difference is that the eigenvalues lie on the unit circle instead of being real. The proofs are also very similar to the Hermitian case.

In the third video we return to rotations on $\mathbb{R}^3$ and study their eigenvalues and eigenvectors. If $R$ describes a rotation by an angle $\theta$ around the axis ${\bf x}$, then the eigenvalues of $R$ are $1$, $e^{i \theta}$ and $e^{-i \theta}$ and ${\bf x}$ is an eigenvector with eigenvalue 1. Since the trace of $R$ is $1 + 2 \cos(\theta)$, we can recover $\theta$ directly from the trace: $$ \theta = \cos^{-1} \left ( \frac{\hbox{(Trace of $R$)}-1}{2} \right ).$$ We go over a few examples to see how it works.

Note that you cannot tell the difference between a clockwise and counterclockwise rotation by just looking at the trace. You have to study the eigenvectors. If ${\bf v}_R + i {\bf v}_I$ is an eigenvector with eigenvalue $e^{i \theta}$, and if the cross product ${\bf v}_I \times {\bf v}_R$ points in the direction of the axis of rotation, then it's a counter-clockwise rotation. If it points in minus the direction of the axis, then it's a clockwise rotation. Note that this also depends on which way you point the axis. A clockwise rotation about the ${\bf x}$ axis is the same thing as a counter-clockwise rotation about the ${-\bf x}$ axis.