Similarity Transformations
A similarity transformation is a linear change of coordinates.
That is, the original -dimensional state vector
is recast
in terms of a new coordinate basis. For any linear
transformation of the coordinate basis, the transformed state vector
may be computed by means of a matrix multiply. Denoting the
matrix of the desired one-to-one linear transformation by
, we
can express the change of coordinates as


Let's now apply the linear transformation to the general
-dimensional state-space description in Eq.
(G.1). Substituting
in Eq.
(G.1) gives
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(G.17) |
Premultiplying the first equation above by

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(G.18) |
Defining
we can write
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(G.20) |
The transformed system describes the same system as in Eq.


![\begin{eqnarray*}
{\tilde H}(z) &=& {\tilde D}+ {\tilde C}(zI - \tilde{A})^{-1}{...
...ht]^{-1} B\\
&=& D + C \left(zI - A\right)^{-1} B\\
&=& H(z)
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/filters/img2172.png)
Since the eigenvalues of are the poles of the system, it follows
that the eigenvalues of
are the same. In other
words, eigenvalues are unaffected by a similarity transformation. We
can easily show this directly: Let
denote an eigenvector of
. Then by definition
, where
is the
eigenvalue corresponding to
. Define
as the
transformed eigenvector. Then we have


The transformed Markov parameters,
, are obviously
the same also since they are given by the inverse
transform of the
transfer function
. However, it is also easy to show this
by direct calculation:

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Modal Representation
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Difference Equations to State Space