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.
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: