Let t be such that ρ$_{1}$ ( t ˆ σ$_{n}$ ) = 1 2 δ$_{1}$ , with 0 . 5 < δ$_{1}$ < δ , and let t$_{1}$ = ε 2 t $_{2}$, then if λ$_{q}$ ( Γ ) ≤ t$_{1}$ we obtain the inequality If λ$_{q}$ ( Γ ) > t$_{1}$ and ‖ B ‖ 2 ≥ t$_{2}$ , all eigenvalues of Γ are smaller than 1 /t q − 1 1 and at least one column of B has a norm greater o equal than t$_{2}$/ √ q . By (A.2), we have and therefore exists a k such that ‖ v$_{k}$ ‖ ≥ t$_{2}$/ √ q where v$_{k}$ is the k th row of V . Then proceeding as in (A.10) we obtain where e$_{k}$ = ( − v$_{k}$ , υ$_{k}$ ) and υ$_{k}$ is the k th row of U $^{′}$. By (A.9), at least nδ values of d$_{i}$ ( B , Γ ) are greater or equal than ε ‖ e$_{k}$ ‖ / √ t q − 1 1 and ‖ e$_{k}$ ‖ 2 = 1 + ‖ v$_{k}$ ‖ 2 ≥ t 2 $_{2}$/q . Then if we take t$_{2}$ = t ε √ q/t q − 1 1 we obtain Then by (A.11) and (A.12) 32 n ∑ i =1 ρ$_{1}$ ( d$_{i}$ ( B , Γ ) ˆ σ$_{n}$ ) ≥ nδρ$_{1}$ ( t ˆ σ$_{n}$ ) = n 2 . (A.12) inf ( B , Γ ) ∈ C n ∑ i =1 ρ$_{1}$ ( d$_{i}$ ( B , Γ ) ˆ σ$_{n}$ ) ≥ n 2 d 2 $_{i}$( B , Γ ) ≥ ( e ′ $_{k}$z$_{i}$ ) $^{2}$t q − 1 1 , ‖ V ‖ 2 = ‖ U $^{′}$B$^{′}$‖ 2 = tr( BUU $^{′}$B$^{′}$) = tr( BB $^{′}$) = ‖ B $^{′}$‖ 2 = ‖ B ‖ 2 $_{2}$, n ∑ i =1 ρ$_{1}$ ( d$_{i}$ ( B , Γ ) ˆ σ$_{n}$ ) ≥ nδρ$_{1}$ ( t ˆ σ$_{n}$ ) > n 2 . (A.11)