I'm a bit confused about what an operator norm is.
Can someone explain it to me using a simple matrix?
What value would you use for the Euclidean norm (x) when it is multiplied by A? (before taking the supremum).
I'm not sure which definition you're using for the operator norm, but the definition I use is if $\displaystyle (X, \|\cdot\|_X), (Y, \|\cdot\|_Y)$ are normed vector spaces and $\displaystyle T : X \to Y$ is linear and bounded, define $\displaystyle \|T\|_{B(X,Y)} := \inf(\{M \in [0,\infty) : \forall x \in X, \|T(x)\|_Y \leq M \|x\|_X\}$. I assume you're using the formula $\displaystyle \|T\|_{B(X,Y)} = \sup(\{ \|T(x)\|_Y : x \in X, \|x\| \leq 1\})$.
Define $\displaystyle A := \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ and let $\displaystyle (x,y)^t \in \mathbb{R}^2$ be given such that $\displaystyle \|(x,y)^t\| \leq 1$. Then $\displaystyle \|A (x,y)^t\| = \|(y, -x)^t\| = \|(x,y)^t\|$. Therefore $\displaystyle \|A\|_{B(X,Y)} = \sup(\{\|(x,y)^t\| : (x,y)^t \in \mathbb{R}^2, \|(x,y)^t\| = 1\}) = 1$.