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*:

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.

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