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 , if we prefer, since the inverse of a one-to-one linear transformation always exists.
Let's now apply the linear transformation
to the general
-dimensional state-space description in Eq.(G.1). Substituting
in Eq.(G.1) gives
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 matrix, and the eigenvalue is unchanged.
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: