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# Lecture 10: Linear Transformations

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Schedule :: Perspective :: Transfer Functions :: Laplace Transform :: Poles and Eigenvalues :: Time Domain :: Linear Transformations :: Special Transformation :: Controllability :: Discrete-Time

For the state equations

let us create a new variable for an invertible matrix . Then , and . Substituting we find

or

where

Similarly,

where

Do these represent the same system?

(Work through details.)

Other observations: eigenvalues? Eigenvectors?

### A special transformation: diagonalizing

Given , suppose that we want to find a transformation matrix such that is diagonal. (This is a convenient form, since it decouples'' all the modes.) How can we find such a ?

Let be eigenvectors of , and be the eigenvalues of , assumed (for our purposes) to be unique. Form

and

where
Then
Identify: , .
Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, June 08). Lecture 10: Linear Transformations. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Signals_and_Systems/10_6node6.html. This work is licensed under a Creative Commons License