Assume we have $A = \begin{bmatrix} A_1 \\ A_2 \end{bmatrix}$, $A$ is $m \times n$ matrix and $A_1$ is non-singular matrix of size $n$ and $$ u = Sup {\{\frac{||y||_2}{||Ay||_2} \quad s.t.\quad y \in C^{n} \quad with \quad y \neq 0}\} $$ $$ v = Sup {\{\frac{||z||_2}{||A_{1}z||_2} \quad s.t.\quad z \in C^{n} \quad with \quad z \neq 0}\} $$ Is it correct to assume $u \leq v$?
I would like to verify whether or not my assumption is right.
Thanks.