  The Single Ring Theorem, by Guionnet, Krishnapur and Zeitouni, describes the
empirical eigenvalue distribution of a large generic matrix with prescribed
singular values, i.e. an $N\times N$ matrix of the form $A=UTV$, with $U, V$
some independent Haar-distributed unitary matrices and $T$ a deterministic
matrix whose singular values are the ones prescribed. In this text, we give a
local version of this result, proving that it remains true at the microscopic
scale $(\log N)^{-1/4}$. On our way to prove it, we prove a matrix
subordination result for singular values of sums of non Hermitian matrices, as
Kargin did for Hermitian matrices. This allows to prove a local law for the
singular values of the sum of two non Hermitian matrices and a delocalization
result for singular vectors.
