Spectral Resolution of the Resolvent of a Matrix

Consider the problem

$$(A-\lambda I) \vec u =\vec b$$

of solving $N$ equations for $N$ unknowns; in other words, given $\vec b$ and the matrix $A$ , find $\vec u$ . Here $\lambda$ is a fixed parameter.

This problem is solved by solving the alternate problem

$$(A-\lambda I) G_\lambda =I$$ (420)

for

$$G_\lambda =(A-\lambda I)^{-1} ~~.$$

The matrix $G_\lambda$ is called the resolvent of the operator $A$ .

The solution $\vec u$ is given by

$$\vec u =G_\lambda \vec b = (A-\lambda I)^{-1} \vec b ~~.$$

This corresponds to expressing the solution in terms of the Green's function.

Continuing with the illustrative example from linear algebra, let us assume that $A$ is Hermitian. Consequently, it has a complete set of eigen vectors $\{ \vec \xi _i:~ i=1, \cdots ,N \}$ , and

$$A = \sum _{i=1}^N \lambda _i \vec \xi _i \vec \xi _i^H$$

$$I = \sum _{i=1}^N ~\vec \xi _i \vec \xi _i^H$$ (421)

$$G_\lambda = \sum _{j=1}^N \frac{ \vec \xi _j \vec \xi _j^H}{\lambda _j -\lambda}$$ (422)

are the spectral representations of $A$ , $I$ , and of the resolvent $G_\lambda =(A-\lambda I)^{-1}$ . The last one follows from the spectral representation

$$A-\lambda I = \sum _{i=1}^N (\lambda _i -\lambda ) \vec \xi _i \vec \xi _i^H$$

and the orthonormality of the eigenvectors:

$$\vec \xi _i^H \vec \xi _j =\delta _{ij}~~.$$

That $G_\lambda$ is the resolvent can be readily checked by verifying Eq.(4.20):

$$(A-\lambda I)G_\lambda = \sum _{i=1}^N \sum _{j=1}^N (\lambda _i -\lambda ) \vec \xi _i \vec \xi _i^H \frac{ \vec \xi _j \vec \xi _j^H}{\lambda _j -\lambda}$$

$$= \sum _{i=1}^N \sum _{j=1}^N \vec \xi _i \delta _{ij} \vec \xi _j^H \frac{\lambda _i -\lambda }{\lambda _j -\lambda}$$

$$= \sum _{i=1}^N \vec \xi _i \vec \xi _i^H$$

$$= I$$

It is clear that the resolvent $G_\lambda$ , Eq.(4.22), viewed as a function of the complex variable $\lambda$ has singularities which are located at $\lambda_1, \cdots, \lambda _N$ , the eigenvalues of $A$ . It follows that a contour integral in the complex $\lambda$ plane around these eigenvalues will recover the spectral representation of the identity, Eq.(4.21). The complex integration leading to this conclusion is straight forward. Consider the integral around the contour $C$ ,

$$\frac{1}{2\pi i} \oint _C G_\lambda ~d\lambda =\frac{1}{2\pi i} \... ... \sum _{j=1}^N \frac{ \vec \xi _j \vec \xi _j^H}{\lambda _j -\lambda} ~d\lambda$$ (423)

Figure 4.6: The integration path in the complex $\lambda$ -plane is large enough so as to enclose all the poles of the resolvent. That integration path is deformed into a union of small circles, each one enclosing a single pole.

Let the closed integration contour $C$ be large enough to enclose all the singularities of $G_\lambda$ . Use the Cauchy-Goursat theorem to deform the closed integration path without changing the value of the integral. Have the deformed integration path consist of the union of cirles, each one enclosing its respective singularity of $G_\lambda$ . This is done in Figure 4.6. As a consequence, the integral over the large contour $C$ becomes a sum of integrals, each one over a small circle surrounding a pole of $G_\lambda$ . Apply Cauchy's integral theorem to each integral. Its value equals the $2\pi i$ times the residue of $G_\lambda$ at that pole, an eigenvalue of $A$ . The residue at the $ith$ eigenvalue is $-\vec \xi _j \vec \xi _j^H$ . Inserting this result into the right hand side of Eq.(4.23), cancelling out the factor $2\pi i$ , one is left with

$$\frac{1}{2\pi i} \oint _C G_\lambda ~d\lambda = -\sum _{j=1}^N \vec \xi _j \vec \xi _j^H$$

This formula is an expression of the following

Proposition: If the contour encloses all singularities of the resolvent $G_\lambda$ in the complex $\lambda$ -plane, then the contour integral yields the complete set of orthonormalized eigenvectors of $A$ or (in view of the completeness of such a set of vectors)

the resolvent of $A$ yields, via complex integration, the spectral representation of the identity,

$$\boxed{ \frac{1}{2\pi i} \oint _C G_\lambda ~d\lambda =-Identity}~~,$$

where

$$\textrm{Identity} =\sum _{j=1}^N \vec \xi _j \vec \xi _j^H ~~.$$

The uniqueness of the resolvent of $A$ guarantees this result no matter how one obtained $G_\lambda$ in the first place. Thus, if one somehow can determine $G_\lambda$ , then by an integration in the complex $\lambda$ -plane one can readily obtain all normalized eigenvectors of the operator $A$ .

Furthermore, it is worth while to emphasize that the nonexistence of $G_\lambda$ for certain values of $\lambda$ , far from being a source of trouble or dispair, is, in fact, an inordinate physical and mathematical boon. As we shall see, from the physical point of view, the nonexistence furnishes us with the resonances of the system, while from the mathematical viewpoint it furnishes us with the orthonormalized eigenvectors or eigenfunctions of the system.

Lecture 33