Skip to content Skip to navigation

Connexions

You are here: Home » Content » The Spectral Representation

Navigation

Recently Viewed

This feature requires Javascript to be enabled.

The Spectral Representation

Module by: Steven J. Cox. E-mail the author

User rating (How does the rating system work?)
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)

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.

With just a little bit more work we shall arrive at a similar expansion for BB itself. We begin by applying the second resolvent identity to Pj Pj . More precisely, we note that the second resolvent identity implies that

B Pj = Pj B=∫ C j zRz−Idz B Pj Pj B z C j z R z I (1)
Pj B=∫ C j zRzdz Pj B z C j z R z Pj B=∫ C j Rzz− λ j dz+ λj ∫ C j Rzdz Pj B z C j R z z λ j λj z C j R z Pj B= Dj + λj Pj Pj B Dj λj Pj Summing this over jj we find
B∑j=1h Pj =∑j=1h λj Pj +∑j=1h Dj B j 1 h Pj j 1 h λj Pj j 1 h Dj (2)
We can go one step further, namely the evaluation of the first sum. This stems from the eqn in the discussion of the transfer function where we integrated Rs Rs over a circle Cρ Cρ where ρ>∥B∥ ρ norm B . The connection to the Pj Pj is made by the residue theorem. More precisely, ∫ Cρ Rzdz=2πⅈ∑j=1h Pj z Cρ R z 2 j 1 h Pj Comparing this to the eqn from the discussion of the transfer function we find
∑j=1h Pj =I j 1 h Pj I (3)
and so Equation 2 takes the form
B=∑j=1h λj Pj +∑j=1h Dj B j 1 h λj Pj j 1 h Dj (4)
It is this formula that we refer to as the Spectral Representation of BB. To the numerous connections between the Pj Pj and Dj Dj we wish to add one more. We first write Equation 1 as B− λj I Pj = Dj B λj I Pj Dj and then raise each side to the mj mj power. As P j m j = Pj P j m j Pj and D j m j =0 D j m j 0 we find
B− λj I m j Pj =0 B λj I m j Pj 0 (5)
For this reason we call the range of Pj Pj the jjth generalized eigenspace, call each of its nonzero members a jjth generalized eigenvector and refer to the dimension of ℛ Pj ℛ Pj as the algebraic multiplicity of λj λj . With regard to the first example from the discussion of the eigenvalue problem, we note that although it has only two linearly independent eigenvectors the span of the associated generalized eigenspaces indeed fills out ℝ3 3 . One may view this as a consequence of P1 + P2 =I P1 P2 I , or, perhaps more concretely, as appending the generalized first eigenvector 010T 010 to the original two eigenvectors 100T 100 and 001T 001 . In still other words, the algebraic multiplicities sum to the ambient dimension (here 3), while the sum of geometric multiplicities falls short (here 2).

Content actions

Give Feedback:

E-mail the module author | Rate 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)

Download:

Module PDF

Add module to:

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 (?)

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