Typical State-Space Diagonalization Procedure

As discussed in [449, p. 362] and exemplified in §C.17.6, to diagonalize a system, we must find the eigenvectors of by solving

for , , where is simply the th pole (eigenvalue of ). The eigenvectors are collected into a similarity transformation matrix:

If there are coupled repeated poles, the corresponding missing eigenvectors can be replaced by generalized eigenvectors.2.12 The matrix is then used to diagonalize the system by means of a simple change of coordinates:

The new diagonalized system is then
 (2.13)

where
 (2.14)

The transformed system describes the same system as in Eq.(1.8) relative to new state-variable coordinates . For example, it can be checked that the transfer-function matrix is unchanged.
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