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

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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University