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The Spectral Representation

Module by: Steven Cox

Summary: (Blank Abstract)

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).

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