16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 8.4.4

Inner Product Spaces

Mathematics — Linear Algebra · unitary-operators-rotation

Problem

Let VV be R2R^2, with the standard inner product. If UU is a unitary operator on VV, show that the matrix of UU in the standard ordered basis is either [cos⁡θ−sin⁡θsin⁡θcos⁡θ]or[cos⁡θsin⁡θsin⁡θ−cos⁡θ]\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 θ\theta, 0≤θ<2π0 \leq \theta < 2\pi. Let UθU_\theta be the linear operator corresponding to the first matrix, i.e., UθU_\theta is rotation through the angle θ\theta. Now convince yourself that every unitary operator on VV is either a rotation, or reflection about the ϵ1\epsilon_1-axis followed by a rotation. (a) What is UθUϕU_\theta U_\phi? (b) Show that Uθ∗=U−θU_\theta^* = U_{-\theta}. (c) Let ϕ\phi be a fixed real number, and let B={α1,α2}\mathfrak{B} = \{\alpha_1, \alpha_2\} be the orthonormal basis obtained by rotating {ϵ1,ϵ2}\{\epsilon_1, \epsilon_2\} through the angle ϕ\phi, i.e., αj=Uϕϵj\alpha_j = U_\phi \epsilon_j. If θ\theta is another real number, what is the matrix of UθU_\theta in the ordered basis B\mathfrak{B}?

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