Why does for every matrix norm $\lVert \mathbb{\cdot }\lVert $
$$\lVert \mathbb{I } \rVert \geq 1$$ hold (where $\mathbb{I }$ is the identity matrix)? I tried to prove it just by the definitions of a matrix norm but I didn't succeed. Can anybody help me?
Matrix norms are usually defined to be submultiplicative (see for example here), that is for any two matrices $A, B$ such that $AB$ exists, we have $$ \def\norm#1{\left\|#1\right\|}\norm{AB} \le \norm A \norm B\tag{$*$} $$ Now, let's look at $\mathrm{id}$, as $\mathrm{id} \ne 0$, we have $\norm{\mathrm{id}} >0$ and, as $\mathrm{id}^2 = \mathrm{id}$ by ($*$), $$ \norm{\mathrm{id}} = \norm{\mathrm{id}^2} \le \norm{\mathrm{id}}^2 $$ Dividing by $\norm{\mathrm{id}}\ne 0$, gives $1 \le \norm{\mathrm{id}}$.