### Properties of the Modal Representation

The vector
in a modal representation (Eq.(G.21)) specifies how
the modes are *driven* by the input. That is, the th mode
receives the input signal weighted by
. In a computational
model of a drum, for example,
may be changed corresponding to
different striking locations on the drumhead.

The vector
in a modal representation (Eq.(G.21)) specifies how
the modes are to be *mixed* into the output. In other words,
specifies how the output signal is to be created as a
*linear combination* of the mode states:

The modal representation is not *unique* since
and
may be scaled in compensating ways to produce the same transfer
function. (The diagonal elements of may also be permuted along
with
and
.) Each element of the state vector
holds the state of a single first-order mode of the system.

For oscillatory systems, the diagonalized state transition matrix must
contain *complex* elements. In particular, if mode is both
oscillatory and *undamped* (lossless), then an excited
state-variable
will oscillate *sinusoidally*,
after the input becomes zero, at some frequency , where

In practice, we often prefer to combine complex-conjugate pole-pairs to form a real, ``block-diagonal'' system; in this case, the transition matrix is block-diagonal with two-by-two real matrices along its diagonal of the form

*complex*multiplies. The function

`cdf2rdf()`in the Matlab Control Toolbox can be used to convert complex diagonal form to real block-diagonal form.

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Jordan Canonical Form

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Example of State-Space Diagonalization