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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

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

 (G.17)

Premultiplying the first equation above by , we have
 (G.18)

Defining
 (G.19)

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

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:

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