## The Jordan Normal Form

The last term of the year has ended and I am already missing the many and varied human connections that I made with my students.  A period of enforced extroversion has ended and I need to cope.  To mute all the emotions of separation I chose the following introvertish task and effectively disappeared into my thoughts.

I had recently reviewed a draft of a student paper on the Jordan Normal Form (JNF), a method of classifying square matrices.  The paper stimulated a desire, even need, to understand JNF’s more deeply.  By understanding I mean building on the models I currently use for linear algebra concepts to the extent that I would be able to explain JNF to other people, namely students. I entered upon this challenge knowing that  a practicing linear algebraist would have a more profound understanding than I could ever have.  The desire to understand and explain somehow touches the core of my being.  I am so lucky to be a teacher of math.

What follows will be fairly technical for what I assume is my usual reader. I will be assuming some familiarity with ideas from linear algebra like eigenvalue, diagonalization and characteristic equation.

After much thinking and rereading my old graduate linear algebra text, Linear Algebra and Matrix Theory by Evar. D. Nering (1963), here is my understanding of the Jordan Normal Form.  JNF is a method of classifying square matrices and therefore linear transformations by their actions on a basis set of vectors for the underlying vector space.  Matrices that have the same actions though on a different set of basis elements are considered similar.

The actions that such linear transformations can have are projection, stretching-shrinking, reflection, rotation and something in my ignorance I will call “shear.”  “Shear” is what you get if a square matrix can not be diagonalized through  standard eigenvalue/eigenvector methods.  There is a particular set of basis vectors (called eigenvectors if the matrix is diagonalizeable) that give a simplified representation of all matrices with a similar action.  If a matrix is not diagonalizeable, there is a clever choice of basis elements that gives a nearly diagonalized form.  This can happen only if there are roots, working in the complex field, of the characteristic equation with multiplicity greater than one.  JNF’s look like this.

Simplified Jordan Normal Matrix Form

At this point after understanding how and why the basis elements are chosen (which I will not discuss here) I got interested in what “shear” was all about, so I wrote a script in Mathematica which I tweaked to produce the following figures annotated using Microsoft OneNote.

First I wanted to see the various actions on the usual orthogonal basis,  in two dimensions.

Projection – Stretch

Stretch-Shrink

Stretch-Reflect

Rotation

Shear

Next I wanted to see what would happen with a nonorthogonal basis like this for example.

2D Nonorthogonal Basis

An exploration of the actions on differing nonorthogonal bases would be interesting but I am stopping with just this one example.

Nonorthogonal Projection – Stretch

Nonorthogonal Stretch-Shrink

Nonorthogonal Stretch-Reflect

Nonorthogonal Rotation

Nonorthogonal Shear

Finally here are some examples from three dimensions with the usual orthogonal basis, .

Projection – Stretch – 3D

Stretch-Shrink-3D

Stretch-Reflect-3D

Shear-3D-Example 1

Shear-3D-Example 2

That’s it.  There are many unanswered questions (effect of different bases, how rotation works, etc) at least for me, but my introvertish self has been temporarily satisfied and now I can move on to preparing for next year.