  We study the $\epsilon$-pseudospectra $\sigma_\epsilon(A)$ of square matrices
$A \in \mathbb{C}^{N \times N}$. We give a complete characterization of the
$\epsilon$-pseudospectrum of any $2 \times 2$ matrix and describe the
asymptotic behavior (as $\epsilon \to 0$) of $\sigma_\epsilon(A)$ for any
square matrix $A$. We also present explicit upper and lower bounds for the
$\epsilon$-pseudospectra of bidiagonal matrices, as well as for finite rank
operators.
