2D Rotation Matrix: Orthogonality and Eigenvalues
Write the matrix $R(\theta)$ that rotates vectors in $\mathbb{R}^2$ counterclockwise by angle $\theta$.
- Verify that $R(\theta)$ is orthogonal (i.e., $R^T R = I$).
- Compute the eigenvalues of $R(\theta)$.
- What does the form of the eigenvalues tell you geometrically about rotations?
Hints
- To build the matrix, think about where the standard basis vectors $e_1 = (1,0)^T$ and $e_2 = (0,1)^T$ land after rotation by $\theta$. Those images become the columns.
- For orthogonality, compute $R^T R$ directly and use the Pythagorean identity $\cos^2\theta + \sin^2\theta = 1$.
- For eigenvalues, solve $\det(R - \lambda I) = 0$. The discriminant will be negative for most $\theta$ -- the eigenvalues are complex conjugates $\cos\theta \pm i\sin\theta$.
Worked Solution
How to Think About It: A rotation in the plane preserves lengths and angles -- it is an isometry. That is exactly what orthogonal matrices do. So before writing anything down, you should expect the matrix to be orthogonal with $|\det R| = 1$. Since a proper rotation preserves orientation, $\det R = +1$. For eigenvalues, think about what rotation does: no real vector in $\mathbb{R}^2$ stays pointing in the same direction after a non-trivial rotation (unless $\theta = 0$ or $\pi$). So the eigenvalues must be complex for generic $\theta$, and they must have modulus 1 (since rotation preserves length).
Quick Sanity Checks: - At $\theta = 0$: $R = I$, eigenvalues $= 1, 1$. Check: $e^{\pm i \cdot 0} = 1$. - At $\theta = \pi/2$: $R$ is a 90-degree rotation, eigenvalues $= \pm i$. Check: $e^{\pm i\pi/2} = \pm i$. - At $\theta = \pi$: $R = -I$, eigenvalues $= -1, -1$. Check: $e^{\pm i\pi} = -1$.
Derivation:
The matrix: To find $R(\theta)$, apply the rotation to the standard basis vectors: - $R(\theta) e_1 = (\cos\theta, \sin\theta)^T$ (the unit vector at angle $\theta$) - $R(\theta) e_2 = (-\sin\theta, \cos\theta)^T$ (rotated by $\theta$ from the $y$-axis)
These are the columns of $R(\theta)$:
$R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$
Orthogonality verification:
$R^T R = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} = \begin{pmatrix} \cos^2\theta + \sin^2\theta & 0 \\ 0 & \cos^2\theta + \sin^2\theta \end{pmatrix} = I$
Also $\det R = \cos^2\theta + \sin^2\theta = 1$, confirming this is a proper rotation (not a reflection).
Eigenvalues:
$\det(R - \lambda I) = (\cos\theta - \lambda)^2 + \sin^2\theta = \lambda^2 - 2\cos\theta \cdot \lambda + 1 = 0$
$\lambda = \frac{2\cos\theta \pm \sqrt{4\cos^2\theta - 4}}{2} = \cos\theta \pm i\sin\theta = e^{\pm i\theta}$
Practical Interpretation: The eigenvalues $e^{\pm i\theta}$ have modulus 1, confirming that rotation preserves vector lengths. The complex form reflects the fact that rotation in $\mathbb{R}^2$ has no real invariant directions (for $\theta \neq 0, \pi$) -- it mixes the two components. The eigenvectors are complex: they correspond to the "circular" basis $(1, i)^T$ and $(1, -i)^T$, which are the left- and right-circular polarization directions.
Answer: The rotation matrix is $R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$. It is orthogonal ($R^T R = I$, $\det R = 1$). Its eigenvalues are $e^{\pm i\theta}$, reflecting that rotation preserves lengths (modulus 1) and has no real invariant directions for generic $\theta$.
Intuition
Rotation matrices are the prototypical example of orthogonal transformations, and understanding their eigenstructure unlocks a lot of linear algebra intuition. The key point: real orthogonal matrices can have complex eigenvalues, and when they do, those eigenvalues always come in conjugate pairs on the unit circle. This is not a pathology -- it is the algebraic encoding of the geometric fact that rotation mixes coordinate directions without stretching.
In quant work, this shows up whenever you diagonalize covariance matrices or work with principal components. The eigenvectors of a real symmetric matrix are always real (spectral theorem), but once you move to non-symmetric operators -- like the dynamics of a mean-reverting process with rotation in phase space -- complex eigenvalues appear and signal oscillatory behavior. Recognizing that $e^{\pm i\theta}$ means "rotation at rate $\theta