8.4: The Spectral Representation
( \newcommand{\kernel}{\mathrm{null}\,}\)
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. More precisely, we note that the second resolvent identity implies that
BPj=PjB
=∫CjzR(z)−Idz
PjB=∫CjzR(z)dz
PjB=∫CjR(z)(z−λj)dz+λj∫CjR(z)dz
PjB=Dj+λjPj
Summing this over j we find
Bh∑j=1Pj=h∑j=1λjPj+h∑j=1Dj
We can go one step further, namely the evaluation of the first sum. This stems from the equation in the discussion of the transfer function where we integrated R(s) over a circle Cρ where ρ>||B||. The connection to the Pj is made by the residue theorem. More precisely,
∫CρR(z)dz=2πih∑j=1Pj
Comparing this to the equation from the discussion of the transfer function we find
h∑j=1Pj=I
and so takes the form
B=h∑j=1λjPj+h∑j=1Dj
It is this formula that we refer to as the Spectral Representation of B. To the numerous connections between the Pj and Dj we wish to add one more. We first write as
(B−λjI)Pj=Dj
and then raise each side to the mj power. As Pmjj=Pj and Dmjj=0 we find
(B−λjI)mPjj=0
For this reason we call the range of Pj the jth generalized eigenspace, call each of its nonzero members a jth generalized eigenvector and refer to the dimension of R(Pj) as the algebraic multiplicity of 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. One may view this as a consequence of P1+P2=I or, perhaps more concretely, as appending the generalized first eigenvector (010)T to the original two eigenvectors (100)T and (001)T. In still other words, the algebraic multiplicities sum to the ambient dimension (here 3), while the sum of geometric multiplicities falls short (here 2).