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We prove and organize some results on the normal forms of Hermitian operatorscomposed with the Veronese map. We apply this general framework to prove twospecific theorems in CR geometry. First, extending a theorem of Faran, weclassify all real-analytic CR maps between any hyperquadric in $\C^2$ and anyhyperquadric in $\C^3$, resulting in a finite list of equivalence classes.Second, we prove that all degree-two CR maps of spheres in all dimensions arespherically equivalent to a monomial map, thus obtaining an elegantclassification of all degree-two CR sphere maps.

Normal forms, Hermitian operators, and CR maps of spheres and hyperquadrics