Spectral radius is not matrix norm. I have seen an example of matrix 
$$A = \begin{bmatrix}
   0 & 1  \\
   0 & 0
  \end{bmatrix}$$
whose spectral radius is zero therefore the spectral radius is not matrix norm. Why the spectral radius is not matrix norm in this case Is it possible that $\|A\|=\epsilon$?

From this Wikipedia page, the spectral norm of a matrix $A\in\mathbb{C}^{n\times n}$ is defined as
$$\rho\left(A\right)=\max_{1\leq i\leq n}\left\{\left|\lambda_{i}\right|\right\}$$
where the $\lambda_{i}$'s are the eigenvalues of the matrix. In your case
$$A=\begin{pmatrix}0&1\\0&0\end{pmatrix}$$
is triangular, so the diagonal entries are the eigenvalues. Thus you have $\lambda_{1}=\lambda_{2}=0$, and it follows immediately that
$$\rho\left(A\right)=0$$
This is not a norm since $A\neq 0$ and a norm $\left\|\cdot\right\|$ must satisfy

$$\left\|A\right\|=0\:\Longleftrightarrow\: A=0$$

by definition.
