content_image/296154.jpg content_image/296155.jpg content_image/296152.jpg content_image/296153.jpg
then if V is positive definite 

<img src='content_image/296154.jpg'>

Remark 6. Recall also that for any two norms || · || a and || · || $_{b}$, we have that 

<img src='content_image/296155.jpg'>

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 $_{q}$. 

<img src='content_image/296152.jpg'>

<img src='content_image/296153.jpg'>

Proof: We only give the main ideas of the proof. Without loss of generality, due 

