## 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

(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):

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