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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

  1. extremely easy (and fast) to calculate
    A v λ v
    just multiply v by λ .
  2. 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 [link] , 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 [link] . 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:

  1. Transform x using V -1 - yields α
  2. Action of A in new basis - a multiplication by Λ
  3. Translate back to old basis - inverse transform using a multiplication by V , which gives us b

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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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