then if V is positive definite Remark 6. Recall also that for any two norms || · || a and || · || $_{b}$, we have that for some α and β and for all matrices A ∈ R r × $^{m}$. In other words, they are equivalent norms, i.e. they induce the same topology in R r × $^{m}$. For || · || a = || · || and || · || b = || · || 2 we have α = 1 and β = √$_{k}$ where k is the rank of the matrix A . Recall also that spectral norm and the Frobenius norm are matrix norms, i.e., for any pair of matrices in R m × m A and B This property will be used in several times. Before proving Theorem 1 we will prove the following Lemma: Lemma 9. Let y ∈ R q and x ∈ R p be fixed vectors. The function is continuous in R p × q × S $_{q}$. Proof: We only give the main ideas of the proof. Without loss of generality, due 29 ‖ AB ‖ 2 ≤ ‖ A ‖ 2 ‖ B ‖ 2 and ‖ AB ‖ ≤ ‖ A ‖‖ B ‖ . (A.4) d $^{2}$( B , Σ ) = ( y − B $^{′}$x ) $^{′}$Σ− $^{1}$( y − B $^{′}$x ) ‖ V ‖ = max j { λ$_{j}$ ( V ) } = λ$_{1}$ ( V ) . (A.3) α ‖ A ‖ a ≤ ‖ A ‖ b ≤ β ‖ A ‖ α