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
(G.17) |
Premultiplying the first equation above by , we have
(G.18) |
Defining
we can write
(G.20) |
The transformed system describes the same system as in Eq.(G.1) relative to new state-variable coordinates. To verify that it's really the same system, from an input/output point of view, let's look at the transfer function using Eq.(G.5):
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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