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. (E.1). Substituting
in Eq. (E.1) gives

(E.17) |

Premultiplying the first equation above by , we have

(E.18) |

Defining

we can write

(E.20) |

The transformed system describes the same system as in Eq. (E.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. (E.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 also by direct calculation:

[How to cite this work] [Order a printed hardcopy]

Copyright ©

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

[Automatic-links disclaimer]