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×n unitary matrices, and elements of the center are diagonal matrices equal to multiplied by the identity matrix. Thus elements of PU(n) correspond to equivalence classes of unitary matrices under multiplication by a constant phase θ.
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).
Read more about Projective Unitary Group: Projective Special Unitary Group, Examples, Finite Fields
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