Interesting examples of non-normal operators? 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)?
 A: 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.
A: Suppose for some vector  $x$ we have $A^2 x = 0$ but $Ax \ne 0$.  Then $A$ is not normal.
In particular, any nonzero nilpotent operator is non-normal.
A: In the finite-dimensional case, spectral theory says that $A$ is normal iff it is diagonalizable and its different eigenspaces are orthogonal to each other.  So you can find non-normal matrices by violating either of these conditions.  For instance, you can just choose any non-orthogonal basis for your inner product space and take a matrix that is diagonal (with distinct diagonal entries) with respect to that basis.  For a non-diagonalizable matrix, take any upper triangular but non-diagonal matrix with constant diagonal entries.
