Let \(V\) be \(\RR^2\), with the standard inner product. If \(U\) is a unitary operator on \(V\), show that the matrix of \(U\) in the standard ordered basis is either \[ \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \quad \text{or} \quad \begin{bmatrix} \cos\theta & \sin\theta \\ \sin\theta & -\cos\theta \end{bmatrix} \] for some real \(\theta\), \(0 \leq \theta < 2\pi\). Let \(U_\theta\) be the linear operator corresponding to the first matrix, i.e., \(U_\theta\) is rotation through the angle \(\theta\). Now convince yourself that every unitary operator on \(V\) is either a rotation, or reflection about the \(\epsilon_1\)-axis followed by a rotation.
What is \(U_\theta U_\phi\)?
Show that \(U_\theta^* = U_{-\theta}\).
Let \(\phi\) be a fixed real number, and let \(\mathfrak{B} = \{\alpha_1, \alpha_2\}\) be the orthonormal basis obtained by rotating \(\{\epsilon_1, \epsilon_2\}\) through the angle \(\phi\), i.e., \(\alpha_j = U_\phi\epsilon_j\). If \(\theta\) is another real number, what is the matrix of \(U_\theta\) in the ordered basis \(\mathfrak{B}\)?
Solution
1. Classify all unitary operators on the real Euclidean plane.
Over \(\mathbb R\), a unitary operator is an orthogonal operator. Let the first column of its standard matrix be \[ \binom ac. \] It has Euclidean norm one, so for a unique \(\theta\in[0,2\pi)\), \[ a=\cos\theta, \qquad c=\sin\theta. \] The second column must be a unit vector perpendicular to the first. There are exactly two such vectors: \[ \binom{-\sin\theta}{\cos\theta} \quad\text{and}\quad \binom{\sin\theta}{-\cos\theta}. \] Thus the matrix is exactly one of \[ \begin{pmatrix} \cos\theta&-\sin\theta\\ \sin\theta&\cos\theta \end{pmatrix}, \qquad \begin{pmatrix} \cos\theta&\sin\theta\\ \sin\theta&-\cos\theta \end{pmatrix}. \]
If \[ R=\begin{pmatrix}1&0\\0&-1\end{pmatrix} \] is reflection in the \(\epsilon_1\)-axis, then the second matrix is \(U_\theta R\). Hence every orthogonal operator is either a rotation or a reflection in the \(\epsilon_1\)-axis followed by a rotation.
2. The product of two rotations adds their angles.
Direct multiplication and the angle-addition identities give \[ \begin{aligned} U_\theta U_\phi &= \begin{pmatrix} \cos\theta&-\sin\theta\\ \sin\theta&\cos\theta \end{pmatrix} \begin{pmatrix} \cos\phi&-\sin\phi\\ \sin\phi&\cos\phi \end{pmatrix}\\ &= \begin{pmatrix} \cos(\theta+\phi)&-\sin(\theta+\phi)\\ \sin(\theta+\phi)&\cos(\theta+\phi) \end{pmatrix}\\ &=U_{\theta+\phi}. \end{aligned} \] Angles may of course be read modulo \(2\pi\).
3. The adjoint of a rotation is rotation through the opposite angle.
Since the standard basis is orthonormal and the matrix is real, \[ [U_\theta^*]=[U_\theta]^t. \] But \[ [U_\theta]^t =\begin{pmatrix} \cos\theta&\sin\theta\\ -\sin\theta&\cos\theta \end{pmatrix} =[U_{-\theta}]. \] Therefore \[ \boxed{U_\theta^*=U_{-\theta}}. \]
4. Rotating the orthonormal basis does not change the matrix of a rotation.
Let \[ \mathfrak B=(U_\phi\epsilon_1,U_\phi\epsilon_2). \] The change-of-basis matrix from \(\mathfrak B\) to the standard basis is \(U_\phi\). Hence \[ [U_\theta]_{\mathfrak B} =U_\phi^{-1}U_\theta U_\phi. \] By steps <1>2–<1>3, \[ U_\phi^{-1}U_\theta U_\phi =U_{-\phi}U_\theta U_\phi =U_{-\phi+\theta+\phi} =U_\theta. \] Thus \[ \boxed{ [U_\theta]_{\mathfrak B} =\begin{pmatrix} \cos\theta&-\sin\theta\\ \sin\theta&\cos\theta \end{pmatrix}.} \]