Interesting examples of non-normal operators?

Question

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)?

Answer 1

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.

Answer 2

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.

Answer 3

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.

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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.