Projective unitary group

In mathematics, the projective unitary group $PU(n)$ is the quotient of the unitary group $U(n)$ by the right multiplication of its center, $U(1)$ , embedded as scalars. Abstractly, it is the holomorphic isometry group of complex projective space, just as the projective orthogonal group is the isometry group of real projective space.

In terms of matrices, elements of $U(n)$ are complex $n \times n$ unitary matrices, and elements of the center are diagonal matrices equal to $e i θ I$ , where $I$ is the identity matrix. Thus, elements of $PU(n)$ correspond to equivalence classes of unitary matrices under multiplication by a constant phase $θ$ . This space is not $SU(n)$ (which only requires the determinant to be one), because $SU(n)$ still contains elements $e i θ I$ where $e i θ$ is an $n$ -th root of unity (since then $det(e i θ I) = e i θn = 1$ ).

Abstractly, given a Hermitian space $V$ , the group $PU(V)$ is the image of the unitary group $U(V)$ in the automorphism group of the projective space $P (V)$ .