# projective unitary group

Basically, the group (algebra) of all unitary matrices where matrices that differ by a multiplicative scaling factor are identified.

That is, if \(A = \lambda B\), then \(A\) and \(B\) are in the same equivalence class.

It turns out that the center \(Z(U(n))\) of the group of all \(n\times n\) unitary matrices \(U(n)\) is exactly the set of scaling matrices \(\lambda I\) for \(\lambda \in \mathbb{C}\).

That is, \(PU(n)\) is the quotient group \(PU(n) = U(n) / Z(U(n))\).

Sources online say that \(Z(U(n))\) is a copy of \(U(1)\). I think this is supposed to mean that it is just \(U(1)\) embedded in a \(n\times n\) matrix, i.e., \(\lambda I\)