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*:

The ones along the superdiagonal serve to couple the states corresponding to and generate polynomial amplitude envelopes multiplying the sampled exponential .

or even three Jordan blocks of order 1. The number of Jordan blocks associated with a single pole 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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University