I was reading this Wikipedia page and found the following confusing:
In that case($A$ begin positive semi-definite), we have $$\left\|I-\frac{A}{\|A\|}\right\|\leq 1$$
It is not clear to me what exactly the above norm is. If it can be an arbitrary matrix norm, then consider the $\ell_1$ norm for $A\in M_{n}$ by $$ \|A\|_1=\sum_{i,j=1}^{n}|a_{ij}| $$ the above inequality will make no sense by direct computation. What is the meaning of the norm here?
I figure that the operator norm makes sense. The reason is as follows:
$A$ being a nonzero positive semi-definite implies that there exists unitary $U$ such that $A=U^\ast\Lambda U$ where $\Lambda$ is a real nonnegative diagonal matrix $\text{diag}\{\lambda_1,\lambda_2,\cdots,\lambda_n\}$, with $\lambda_1\ge\lambda_2\ge\cdots\lambda_n\ge 0$ and $\lambda_1>0$. In a Hilbert space, we have $$\left\|I-\frac{A}{\|A\|}\right\|=\left\|I-\frac{\Lambda}{\|\Lambda\|}\right\|=\left\|I-\frac{\Lambda}{\lambda_1}\right\|\le 1.$$