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Jones calculus represents a fully polarized monochromatic light field by a two-component Jones vector of transverse complex amplitudes. An ideal linear optical element acts by a complex $2\times 2$ matrix; successive elements act by the ordered product of their matrices. The order matters because the transformations need not commute.

For transverse axes $x,y$, an ideal polarizer transmitting $x$ has matrix $P=\mathrm{diag}(1,0)$, so $P(E_x,E_y{)}^{\mathsf{T}}=(E_x,0{)}^{\mathsf{T}}$. A quarter-wave plate aligned with those axes can be represented, up to a common phase factor and the chosen sign convention, by $Q=\mathrm{diag}(1,i)$. It changes the relative wave phase, enabling conversion between linear polarization and elliptical polarization. Rotating the elements tests these predictions by changes in transmitted light intensity. A single Jones vector does not describe arbitrary partially polarized light; that requires statistics of the field rather than one fixed pair of amplitudes.