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
or
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) |
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(G.18) |
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(G.20) |
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
Thus, the transformed eigenvector is an eigenvector of the transformed
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: