Skip to content Skip to navigation

Connexions

You are here: Home » Content » The Eigenvalue Problem

Navigation

Content Actions

  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of a lens

      Lenses

      A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

      What is in a lens?

      Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

      Who can create a lens?

      Any individual Connexions member, a community, or a respected organization.

      What are tags? tag icon

      Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

    • External bookmarks
  • E-mail the author
  • Rate this module (How does the rating system work?)

    Rating system

    Ratings

    Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

    How to rate a module

    Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

    (0 ratings)

Recently Viewed

This feature requires Javascript to be enabled.

The Eigenvalue Problem

Module by: Steven Cox

Summary: (Blank Abstract)

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

Introduction

Harking back to our previous discussion of The Laplace Transform we labeled the complex number λλ an eigenvalue of BB if λIB λ I B was not invertible. In order to find such λλ one has only to find those ss for which sIB-1 s I B is not defined. To take a concrete example we note that if

B=110010002 B 110 010 002 (1)
then
sIB-1=1s12s2s1s2s200s1s2000s12 s I B 1 s 1 2 s 2 s1 s2 s2 0 0 s1 s2 0 0 0 s1 2 (2)
and so λ1=1 λ1 1 and λ2=2 λ2 2 are the two eigenvalues of BB. Now, to say that λj IB λj I B is not invertible is to say that its columns are linearly dependent, or, equivalently, that the null space 𝒩 λj IB 𝒩 λj I B contains more than just the zero vector. We call 𝒩 λj IB 𝒩 λj I B the jjth eigenspace and call each of its nonzero members a jjth eigenvector. The dimension of 𝒩 λj IB 𝒩 λj I B is referred to as the geometric multiplicity of λj λj . With respect to BB above, we compute 𝒩 λ1 IB 𝒩 λ1 I B by solving IBx=0 IB x 0 , i.e., 0-10000001x1x2x3=000 0 -1 0 000 001 x1 x2 x3 0 0 0 Clearly 𝒩 λ1 IB={c100T|c} 𝒩 λ1 I B c 100 c Arguing along the same lines we also find 𝒩 λ2 IB={c001T|c} 𝒩 λ2 I B c 001 c That BB is 3x3 but possesses only 2 linearly eigenvectors leads us to speak of BB as defective. The cause of its defect is most likely the fact that λ1 λ1 is a double pole of sIB-1 sI B In order to flesh out that remark and uncover the missing eigenvector we must take a much closer look at the transfer function RssIB-1 R s sI B In the mathematical literature this quantity is typically referred to as the Resolvent of BB.

Comments, questions, feedback, criticisms?

Send feedback