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Nonzero scalar matrices form a subgroup $N$ of the general linear group, since products and inverses remain nonzero scalar matrices. Moreover, for every invertible $g$,Thus $N$ lies in the center of a group and is a normal subgroup. The quotient group is the complex projective linear group ${\mathrm{PGL}}_n(\mathbb{C})$, whose elements identify invertible matrices differing by a nonzero scalar.

The canonical group homomorphism $\pi :g\mapsto [g]$ obeys $[g][h]=[gh]$. Composing it with the given group representation givesConsequentlyThis is the projective representation associated with $R$: a nonzero scalar acts trivially on each line in the projective space of $V$. Its kernel of a group homomorphism consists of those $g$ for which $R(g)$ is scalar, which can be larger than the kernel of a group homomorphism of $R$ itself.

Let $A=\mathrm{diag}(1,-1)$ and $B=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$. They generate the dihedral group of a square, realized by $\{\pm I,\pm A,\pm B,\pm AB\}$. Its image in the projective linear group has four elements, since opposite matrices become equal. The projective classes of $A,B$ commute because $AB=-BA$, and each squares to the identity. Sending $[A^a{B}^b]$ to $(a,b)$ gives the Klein four-group. This is an example in which a nontrivial scalar commutator disappears after passing to a projective representation.

A projective representation is a group homomorphism into the projective linear group of a vector space. Choosing invertible linear representatives makes multiplication hold up to nonzero scalar factors. A projective unitary representation chooses unitary operators as representatives, whose scalar factors have modulus one.

Because state vectors differing by a quantum phase represent the same ray, a unitary realization of a group action need only satisfy

A projective unitary representation is a projective representation whose chosen representatives are unitary operators. The phase factors disappear on rays.