SL(n,C)

Decomposition into unitary and Hermitian parts

Any SL(n,C) matrix $M\,$ can be uniquely written as

$M = V \rho\,$

where

$V^\dagger V = 1\,$ ,	$\det V = 1\,$ ,	i.e., $V \in SU(n)\,$
$\rho^\dagger = \rho\,$ ,	$\det \rho = 1\,$ ,	i.e., $\rho\,$ is Hermitian.

Note that both $\rho, V \in SL(n,C)\,$ . We can diagonalize $M^\dagger M\,$ since it is Hermitian matrix (self-adjoint). Let $U\,$ be a unitary matrix that diagonalizes $M^\dagger M\,$ .

$U M^\dagger M U^\dagger = D\,$ .

Then, define

$\rho \equiv U^\dagger \sqrt{D} U\,$ .

Since $\det M = 1\,$ , $M\,$ has no zero eigenvalues, so that $\rho\,$ is invertible. Therefore we can define the matrix

$V \equiv M \rho^{-1}\,$ .

It also follows that

$\rho^\dagger \rho = M^\dagger M\,$ .

Since $\rho\,$ is obtained from $M^\dagger M\,$ , the matrix $M' = g M\,$ , $g \in U(n)\,$ results in the same $\rho\,$ .