### 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*:

^{G.11}Note, however, that a pole of multiplicity three can also yield two Jordan blocks, such as

*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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