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A Jones vector records the two transverse complex amplitudes of a completely polarized monochromatic wave. Multiplication by a common nonzero complex number preserves its polarization of waves, while changing overall amplitude and phase. With the convention $\mathbf{\Psi }(t)=\mathrm{Re}[\mathbf{J}{e}^{i\omega t}]$, $\mathbf{J}=(1,2i{)}^{\mathsf{T}}$ gives $(\cos \omega t,-2\sin \omega t)$, an ellipse with the longer axis along $y$. Handedness labels require both a viewing direction and a naming convention.

A Jones matrix is a complex two-by-two linear transformation of a Jones vector through a deterministic nondepolarizing optical element. Successive elements multiply in reverse reading order. An ideal linear polarizer along a real transverse unit vector $\mathbf{a}$ has matrix $\mathbf{a}{\mathbf{a}}^{\mathsf{T}}$. A common nonzero scalar phase does not change the output polarization of waves.