Here is a puzzle concerning the optics of polarized light. Every state of polarization has its opposite. For linear polarization, it's linear but rotated 90 degrees. For circular polarization, it's circular with opposite helicity. For general elliptic polarization, it's elliptic with the ellipse rotated 90 degrees, and the helicity reversed. On the Poincaré sphere, opposite states of polarization are represented by diametrically opposite points. The puzzle is to construct an optical device that reverses the polarization state. For any input, the output is the opposite polarization. Or, prove that it can't be done. I need to describe some of the machinery of polarization optics. The electric field of a plane wave propagating in the z direction is x component: Re(Ex exp(i(kz - omega t)), y component: Re(Ey exp(i(kz - omega t)). Ex and Ey are complex, encoding a magnitude and phase shift. It is crucial to the proof that these amplitudes are complex. The input and output beams may be in different directions. We pick coordinates in the transverse planes, (x,y) for the input beam, (x',y') for the output beam. The state of polarization is specified by the ratio P = Ey/Ex. If P is real, the x and y components are in phase, and the light is linearly polarized. If P = +-i, the components are equal in magnitude and 90 degrees out of phase, and the light is circularly polarized. Topologically, the values P form a sphere, the Poincaré sphere. When presenting my puzzle to optical science people, it's convenient to use the Poincaré sphere to make precise the meaning of "opposite", but I don't need to make use of it here. The output amplitudes are related to the input amplitudes by a linear transformation. [ Ex' ] = [ Jxx Jxy ] [ Ex ][ Ey' ] [ Jyx Jyy ] [ Ey ]. This matrix J is known as the Jones matrix. My Jones matrix must take every polarization state P to its opposite P'. I don't need to say exactly what "opposite" means. The only property I need is that no state is its own opposite. An eigenvector of J satisfies Ey'/Ex' = Ey/Ex. A polarization state P is preserved by J if and only if (Ex,Ey) with P = Ey/Ex is an eigenvector. But my J preserves no polarization state; it has no eigenvectors. This contradicts a theorem of linear algebra that every linear operator on a finite dimensional vector space over an algebraically closed field has at least one eigenvector. It follows that every optical system must preserve at least one polarization state. -- Gene