Eigen-stuff in a Nutshell 2.3 2002/07/17 2003/07/24 10:54:43 GMT-5 Michael Haag mjhaag@rice.edu Justin Romberg jrom@rice.edu Justin Romberg jrom@rice.edu Richard G. Baraniuk richb@rice.edu Michael Haag mjhaag@rice.edu Mariyah Poonawala mariyah@rice.edu Prashant Singh prash@ece.rice.edu eigen eigenfunction eigenfunctions eigenvalue eigenvalues eigenvector eigenvectors LTI span system This module provides a brief overview and review of the importance of eigenvectors and eigenvalues in analyzing and understanding LTI systems.

A Matrix and its Eigenvector The reason we are stressing eigenvectors and their importance is because the action of a matrix A on one of its eigenvectors v is extremely easy (and fast) to calculate A v λ v just multiply v by λ . easy to interpret: A just scales v, keeping its direction constant and only altering the vector's length. If only every vector were an eigenvector of A....

Using Eigenvectors' Span Of course, not every vector can be ... BUT ... For certain matrices (including ones with distinct eigenvalues, λ's), their eigenvectors span n , meaning that for any x n , we can find α 1 α 2 α n such that: x α 1 v 1 α 2 v 2 … α n v n Given , we can rewrite A x b . This equation is modeled in our LTI system pictured below:

LTI System.

x i α i v i b i α i λ i v i The LTI system above represents our . Below is an illustration of the steps taken to go from x to b. x α V -1 x Λ V -1 x V Λ V -1 x b where the three steps (arrows) in the above illustration represent the following three operations: Transform x using V -1 - yields α Action of A in new basis - a multiplication by Λ Translate back to old basis - inverse transform using a multiplication by V , which gives us b