# Lecture 10: Linear Transformations

For the state equations

or

where

Similarly,

where

Instead of we have .

Do these represent the same system?

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