## Repeated Poles

The above summary of state-space diagonalization works as stated when the modes (poles) of the system are distinct. When there are two or more resonant modes corresponding to the same ``natural frequency'' (eigenvalue of ), then there are two further subcases: If the eigenvectors corresponding to the repeated eigenvalue (pole) are*linearly independent*, then the modes are independent and can be treated as distinct (the system can be diagonalized). Otherwise, we say the equal modes are

*coupled*.

The coupled-repeated-poles situation is detected when the matrix of eigenvectors

`V`returned by the

`eig`matlab function [

*e.g.*, by saying

`[V,D] = eig(A)`] turns out to be

*singular*. Singularity of

`V`can be defined as when its

*condition number*[

`cond(V)`] exceeds some threshold, such as

`1E7`. In this case, the linearly dependent eigenvectors can be replaced by so-called

*generalized eigenvectors*[58]. Use of that similarity transformation then produces a ``block diagonalized'' system instead of a diagonalized system, and one of the blocks along the diagonal will be a matrix corresponding to the pole repeated times. Connecting with the discussion regarding repeated poles in §6.8.5, the Jordan block corresponding to a pole repeated times plays exactly the same role of repeated poles encountered in a partial-fraction expansion, giving rise to terms in the impulse response proportional to , , and so on, up to , where denotes the repeated pole itself (

*i.e.*, the repeated eigenvalue of the state-transition matrix ).

### 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.

**Next Section:**

State-Space Analysis Example: The Digital Waveguide Oscillator

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Modal Representation