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Eigen-stuff in a Nutshell

Module by: Michael Haag, Justin Romberg

Summary: 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 AA on one of its eigenvectors vv is
  1. extremely easy (and fast) to calculate
    Av=λv A v λ v (1)
    just multiply vv by λ λ.
  2. easy to interpret: AA just scales vv, keeping its direction constant and only altering the vector's length.
If only every vector were an eigenvector of AA....

Using Eigenvectors' Span

Of course, not every vector can be ... BUT ... For certain matrices (including ones with distinct eigenvalues, λλ's), their eigenvectors span ℂn n , meaning that for any x∈ℂn x n , we can find α 1 α 2 α n ∈ℂ α 1 α 2 α n such that:
x= α 1 v1+ α 2 v2+…+ α n vn x α 1 v 1 α 2 v 2 … α n v n (2)
Given Equation 2, we can rewrite Ax=b A x b . This equation is modeled in our LTI system pictured below:
eigv_sys.png
Figure 1: LTI System.
x=∑i α i vi x i α i v i b=∑i α i λ i vi b i α i λ i v i The LTI system above represents our Equation 1. Below is an illustration of the steps taken to go from xx to bb. x→α=V-1x→ΛV-1x→VΛV-1x=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:
  1. Transform x x using V-1 V -1 - yields αα
  2. Action of AA in new basis - a multiplication by Λ Λ
  3. Translate back to old basis - inverse transform using a multiplication by V V, which gives us bb

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