### 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:

*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

*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