JordanForm

JordanForm.jl is an implementation for computing the Jordan form, also known as Jordan normal form and Jordan canonical form, intended for educational use. The Jordan form $J$ is a useful almost diagonal structure that is always computable, even if a matrix $A$ is not diagonaliziable. It is commonly used in the study of linear algebra and differential equations. In particular, if we consider a linear system of differential equations $\dot{x} = Ax$, then the Jordan form of $A$ can be used to find the general solution $x(t) = e^{At}x_0 = Se^{Jt}S^{-1}x_0$ of the system where $S$ is the transformation matrix between $A$ and $J$. That is, $AS = SJ$. Furthermore, there exists an analytic formula for $e^{Jt}$. The transformation matrix $S$ is computed as the concatenation of the generalized eigenvectors of $A$.

However, the structure of $J$ is very unstable under perturbation, as it depends on repeated eigenvalues of $A$. Therefore, it is not recommended to use the Jordan form for numerical computations. So why did we write a library to compute the Jordan form? The Jordan form is a useful tool for studying and understanding matrices and linear systems - and it is fun to write code!

You can read more about the Jordan form on Wikipedia.

Warning

The Jordan form is not numerically stable. Attempting to use this with floating point numbers may yield unexpected results. This package is intended for educational use only.

Usage

As previously noted, the Jordan form is not numerically stable. Therefore, we restrict the input to integer and rational matrices. The Jordan form of a matrix $A$ is computed using the function jordan_form(A). The function returns a factorization struct F = (S, J) where J is the Jordan form of A and S is the transformation matrix such that A * S = S * J.

using JordanForm 

A = [
    1 1 0;
    0 2 1;
    0 0 3
]

F = jordan_form(A)  # or
S, J = jordan_form(A)

@assert A * S ≈ S * J