trying to understand the topic of 'A minimal norm completion problem'. There's an exercise I'm not able to solve yet.
Let $A \in \mathbb C^{p\times q}$. Then, $$ ||A|| \le \gamma \iff \gamma^2I_q - A^HA \ge 0\iff \gamma^2I_p - AA^H \ge 0 \quad (1)$$
I bet this is easily extracted from the inequalities that I proved in this exercise: (by using SVD of A) $$ ||A|| \le 1 \iff I_q - A^HA \ge 0\iff I_p - AA^H \ge 0 \quad (2)$$
I'm not sure how can I generalize on $\gamma$, can anyone show me what I'm missing in order to satisify the inequalities in $(1)$?
So we know $\forall A\in\mathbb{C}^{p\times q}$: $$ \Vert A\Vert\leq 1\iff I_q-A^HA\geq 0\iff I_p-AA^H\geq 0 $$ If $\Vert B\Vert \leq\gamma$, then $\Vert\gamma^{-1}B\Vert\leq 1$ and we can apply the above statement: $$ \Vert \gamma^{-1}B\Vert\leq 1\iff I_q-(\gamma^{-1}B)^H(\gamma^{-1}B)\geq 0\iff I_p-(\gamma^{-1}B)(\gamma^{-1}B)^H\geq 0 $$ But every one of these respective statements is equivalent to the corresponding statement in $$ \Vert B\Vert\leq \gamma\iff \gamma^2I_q-B^HB\geq 0\iff \gamma^2I_p-BB^H\geq 0 $$ Thus $(1)$ is indeed a direct consequence of $(2)$.