A vector, say for example $ a = \begin{bmatrix} 1 \\ 2\end{bmatrix}$, can be regarded not only as a vector but also as a 2 x 1 matrix. If we interpret it as a vector we can compute the $\ell_p$ norm of $a$. If we interpret it as a matrix we can compute its matrix norm, i.e:
$$ ||a|| = \max_{x \neq 0} \frac{||ax||}{||x||} $$
My question is: If we interpret $a$ as a matrix, is the matrix norm equal to any of the vector norms? I suspect that it would equal the Euclidean norm, as using the Cauchy–Schwarz inequality yields
$$ ||a|| = \max_{x \neq 0} \frac{||ax||}{||x||} \leq \frac{||a|| ||x||}{||x||} = ||a|| = \sqrt5 $$
but I am not sure if this is correct. Any help would me highly appreciated! Thanks