TITLE: Show that the identity matrix $I$ must have norm $1$.
QUESTION [9 upvotes]: I am trying to understand why the identity matrix $I$ must have a norm $1$, for any choice of matrix-norm $|\cdot|$?  How would i show this?

REPLY [7 votes]: As many others have pointed out, the matrix norm of $I$ is not necessarily equal to $1$. In fact, if $\|\cdot\|$ is a matrix norm $c\|\cdot\|$ is also a matrix norm for any $c\ge1$.
Yet, if your matrix norm is submultiplicative, we have $0<\|I\|=\|I^2\|\le\|I\|^2$ and hence $\|I\|\ge1$, but strict inequality may still occur. E.g. the Frobenius norm $\|\cdot\|_F$ is submultiplicative, but $\|I_n\|_F=\sqrt{n}>1$ when $n>1$.
If your textbook is not erred, it is perhaps talking about an induced matrix norm, i.e. one defined by $\|A\|=\sup_{x\neq0}\frac{\color{red}{\|}Ax\color{red}{\|}}{\color{red}{\|}x\color{red}{\|}}$ for some vector norm $\color{red}{\|\cdot\|}$ defined on $K^n$. In this case, it is straight from the definition that $\|I\|=1$.