8.4: The Spectral Representation
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 \(P_{j}\). More precisely, we note that the second resolvent identity implies that
\[BP_{j}=P_{j}B \nonumber\]
\[= \int_{C_{j}} zR(z)-I dz \nonumber\]
\[P_{j}B = \int_{C_{j}} zR(z) dz \nonumber\]
\[P_{j}B = \int_{C_{j}} R(z)(z-\lambda_{j})dz+\lambda_{j} \int_{C_{j}} R(z) dz \nonumber\]
\[P_{j}B = D_{j}+\lambda_{j} P_{j} \nonumber\]
Summing this over \(j\) we find
\[B \sum_{j = 1}^{h} Pj= \sum_{j = 1}^{h} \lambda_{j} P_{j}+ \sum_{j = 1}^{h} D_{j} \nonumber\]
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 \(C_{\rho}\) where \(\rho > ||B||\). The connection to the \(P_{j}\) is made by the residue theorem. More precisely,
\[\int_{C_{\rho}} R(z) dz = 2\pi i \sum_{j=1}^{h} P_{j} \nonumber\]
Comparing this to the equation from the discussion of the transfer function we find
\[\sum_{j=1}^{h} P_{j} = I \nonumber\]
and so takes the form
\[B = \sum_{j=1}^{h} \lambda_{j} P_{j}+ \sum_{j=1}^{h} D_{j} \nonumber\]
It is this formula that we refer to as the Spectral Representation of \(B\). To the numerous connections between the \(P_{j}\) and \(D_{j}\) we wish to add one more. We first write as
\[(B-\lambda_{j} I)P_{j} = D_{j} \nonumber\]
and then raise each side to the \(m_{j}\) power. As \(P_{j}^{m_{j}} = P_{j}\) and \(D_{j}^{m_{j}} = 0\) we find
\[(B-\lambda_{j}I)m_{j}^{P_{j}} = 0 \nonumber\]
For this reason we call the range of \(P_{j}\) the jth generalized eigenspace , call each of its nonzero members a jth generalized eigenvector and refer to the dimension of \(\mathscr{R}(P_{j})\) as the algebraic multiplicity of \(\lambda_{j}\). 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 \(\mathbb{R}^{3}\). One may view this as a consequence of \(P_{1}+P_{2} = I\) or, perhaps more concretely, as appending the generalized first eigenvector \(\begin{pmatrix} {0}&{1}&{0} \end{pmatrix}^T\) to the original two eigenvectors \(\begin{pmatrix} {1}&{0}&{0} \end{pmatrix}^T\) and \(\begin{pmatrix} {0}&{0}&{1} \end{pmatrix}^T\). In still other words, the algebraic multiplicities sum to the ambient dimension (here 3), while the sum of geometric multiplicities falls short (here 2).