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 
of multiplicity 
could give
rise to the following 
Jordan block:
![$\displaystyle D_i = \left[\begin{array}{ccc}
p_i & 1 & 0\\ [2pt]
0 & p_i & 1\\ [2pt]
0 & 0 & p_i
\end{array}\right]
$](http://www.dsprelated.com/josimages_new/filters/img2223.png)
and generate polynomial amplitude envelopes
multiplying the sampled exponential 
.G.11Note, however, that a
pole of multiplicity three can also yield two Jordan blocks, such as
![$\displaystyle D_i = \left[\begin{array}{ccc}
p_i & 1 & 0\\ [2pt]
0 & p_i & 0\\ [2pt]
0 & 0 & p_i
\end{array}\right]
$](http://www.dsprelated.com/josimages_new/filters/img2236.png)
is equal to the number of linearly
independent eigenvectors of the transition matrix associated with
eigenvalue . If all eigenvectors of 
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 toolbox. 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.
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