# Why does for every matrix norm $\lVert \mathbb{I } \rVert \geq 1$ hold?

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?

• $\|A B\|\le \|A\|\ \|B\|$?
– user99914
Oct 11, 2015 at 23:23
• I was going to say, they probably assume only sub-multiplicative norms. Oct 11, 2015 at 23:27
• $A = \mathbb{I}$ so, $\lVert \mathbb{I} \rVert \lVert B \rVert \geq \lVert \mathbb{I } B \rVert$. What is B? Oct 11, 2015 at 23:34
• $B = A$... ${}{}{}$
– user99914
Oct 12, 2015 at 2:17

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 $1 \le \norm{\mathrm{id}}$.