This article is a very good introduction to Jordan matrices and the normal form, but it skips a few steps which I think the authors thought were obvious, but may not be.
I fill in a few steps here.
The definition of Jordan normal form (JNF) is tricky.
The article is clear but I reiterate here:
JNF is useful to learn about the pathologies of eigenvalues and eigenvectors.
- A matrix is in JNF iff this says so.
- M is in JNF iff:
- Mj, j+1 is 0 or 1,
- If Mj, j+1 = 1 then Mj, j = Mj+1, j+1,
- If j<i or i+1<j then Mi, j = 0.
The defining equation for eigenvalues and eigenvectors of matrix M is Mx = λx iff x is an eigenvector and λ is the corresponding eigenvalue.
λ is some field member; here we take it to be a complex number.
We first view M as a linear transformation of a vector space to itself—no basis mentioned yet.
The vector space is over the complex numbers too.
It should be noted that no quadratic form is in sight.
When we choose some specific basis for the vector space we can examine M as a matrix of complex numbers.
As the article says for an arbitrary transformation there is some choice of basis so that its matrix M is in JNF.
By studying the eigenvalues and eigenvectors of JNF matrices we learn about what can go wrong.
Furthermore we can produce a non JNF matrix B with arbitrary pathology by starting with a JNF matrix J, choosing a random invertible matrix A and writing B = A−1JA.
This would seem to be the ultimate stress test for eigenvector routines.
Consider this matrix:
According to the text of the article all eigenvalues are 2 and there is only one eigenvector: (1, 0, 0).
Its characteristic polynomial is (λ — 2)3.
This works for any complex number in place of 2.
Actually you can convince yourself of this by as especially simple exercise in solving for the three unknowns components of the vector.