Jordan Canonical Form

The block diagonal system having the eigenvalues along the diagonal and ones in some of the superdiagonal elements (which serve to couple repeated eigenvalues) is called Jordan canonical form. Each block size corresponds to the multiplicity of the repeated pole. As an example, a pole $p_i$ of multiplicity $m_i=3$ could give rise to the following $3\times 3$ Jordan block:

$$D_i = \left[\begin{array}{ccc} p_i & 1 & 0\\ [2pt] 0 & p_i & 1\\ [2pt] 0 & 0 & p_i \end{array}\right]$$

The ones along the superdiagonal serve to couple the states corresponding to $p_i$ and generate polynomial amplitude envelopes multiplying the sampled exponential $p_i^n$.^E.11Note, however, that a pole of multiplicity three can also yield two Jordan blocks, such as

$$D_i = \left[\begin{array}{ccc} p_i & 1 & 0\\ [2pt] 0 & p_i & 0\\ [2pt] 0 & 0 & p_i \end{array}\right]$$

or even three Jordan blocks of order 1. The number of Jordan blocks associated with a single pole $p_i$ is equal to the number of linearly independent eigenvectors of the transition matrix associated with eigenvalue $p_i$. If all eigenvectors of $A$ are linearly independent, the system can be diagonalized after all, and any repeated roots are ``uncoupled'' and behave like non-repeated roots (no polynomial amplitude envelopes).

Interestingly, neither Matlab nor Octave seem to have a numerical function for computing the Jordan canonical form of a matrix. Matlab will try to do it symbolically when the matrix entries are given as exact rational numbers (ratios of integers) by the jordan function, which requires the Maple symbolic mathematics tool box. Numerically, it is generally difficult to distinguish between poles that are repeated exactly, and poles that are merely close together. The residuez function sets a numerical threshold below which poles are treated as repeated.