Eigenvalue Problem: The Transfer Function 2.2 2002/02/04 2002/07/19 Steven Cox cox@rice.edu Charlet Reedstrom charlet@rice.edu Jacob Grabczewski jgrab@owlnet.rice.edu Steven Cox cox@rice.edu eigenvalue transfer function (Blank Abstract)

The Transfer Function One means by which to come to grips with R s is to treat it as the matrix analog of the scalar function 1 sb This function is a scaled version of the even simpler function 1 1z . This latter function satisfies the identity (just multiply across by 1z to check it) 1 1z 1 z z2 ... z n1 zn 1z for each positive integer n. Furthermore, if z 1 then z n 0 as n and so becomes, in the limit, 1 1z n 0 zn the familiar geometric series. Returning to we write 1 sb 1 s 1 bs 1s b s 2 ... b n1 sn b n s n 1 sb and hence, so long as s b we find, 1 sb 1 s n 0 b s n This same line of reasoning may be applied in the matrix case. That is, s I B s I Bs 1 s B s2 ... B n1 sn Bn sn sI B and hence, so long as s B where B is the magnitude of the largest element of B, we find sI B s n 0 B s n Although is indeed a formula for the transfer function you may, regarding computation, not find it any more attractive than the Gauss-Jordan method. We view however as an analytical rather than computational tool. More precisely, it facilitates the computation of integrals of R s . For example, if Cρ is the circle of radius ρ centered at the origin and ρ B then s Cρ sI B n 0 B n s Cρ s -1n 2 I This result is essential to our study of the eigenvalue problem. As are the two resolvent identities. Regarding the first we deduce from the simple observation s2 I B s1 I B s2 I B s1 I B s2 I B s1 I B that R s2 R s1 s1 s2 R s2 R s1 The second identity is simply a rewriting of sI B sI B sI B sI B I namely, B R s R s B s Rs I