TITLE: Interesting examples of non-normal operators?
QUESTION [5 upvotes]: I am currently learning spectral aspects of linear algebra.
At first sight, it seems like normality is very narrow restriction. But, I can not think up any examples of non-normal operators.
There is example on Wiki, but it is just matrix, and I want to find out some broad and intuitive class of examples. 
What are examples of non-normal operators (especially finite-dimensional)?

REPLY [4 votes]: Here is a helpful bit of knowledge: an upper-triangular matrix is normal if and only if it is diagonal.
By the Schur triangularization theorem, every matrix is unitarily similar to an upper-triangular matrix.  So, up to unitary similarity, every normal matrix diagonal, and every non-normal matrix is upper-triangular, but not diagonal.
So, in particular, matrices of the form
$$
\pmatrix{\lambda_1 & 1\\0 & \lambda_2}
$$
are never normal.

An interesting infinite dimensional example: a linear isometry is unitary if and only if it is normal, which is true if and only if it is surjective.