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Eigenvalue Problem: The Transfer Function

Module by: Steven Cox

Summary: (Blank Abstract)

The Transfer Function

One means by which to come to grips with Rs R s is to treat it as the matrix analog of the scalar function

1s-b 1 sb (1)
This function is a scaled version of the even simpler function 11-z 1 1z . This latter function satisfies the identity (just multiply across by 1-z1z to check it)
11-z=1+z+z2+...+zn-1+zn1-z 1 1z 1 z z2 ... z n1 zn 1z (2)
for each positive integer nn. Furthermore, if |z|<1 z 1 then zn→0 z n 0 as n→∞ n and so Equation 2 becomes, in the limit, 11-z=∑n=0∞zn 1 1z n 0 zn the familiar geometric series. Returning to Equation 1 we write 1s-b=1s1-bs=1s+bs2+...+bn-1sn+bnsn1s-b 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| s b we find, 1s-b=1s∑n=0∞bsn 1 sb 1 s n 0 b s n This same line of reasoning may be applied in the matrix case. That is,
sI-B-1=s-1I-Bs-11s+Bs2+...+Bn-1sn+BnsnsI-B-1 s I B s I Bs 1 s B s2 ... B n1 sn Bn sn sI B (3)
and hence, so long as |s|>∥B∥ s B where ∥B∥ B is the magnitude of the largest element of BB, we find
sI-B-1=s-1∑n=0∞Bsn sI B s n 0 B s n (4)
Although Equation 4 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 Equation 4 however as an analytical rather than computational tool. More precisely, it facilitates the computation of integrals of Rs R s . For example, if Cρ Cρ is the circle of radius ρρ centered at the origin and ρ>∥B∥ ρ B then
∫ Cρ sI-B-1ds=∑n=0∞Bn∫ Cρ s-1-nds=2πⅈI s Cρ sI B n 0 B n s Cρ s -1n 2 I (5)
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-1- s1 I-B-1= s2 I-B-1 s1 I-B- s2 I+B s1 I-B-1 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 R s2 R s1 s1 s2 R s2 R s1 (6)
The second identity is simply a rewriting of sI-BsI-B-1=sI-B-1sI-B=I sI B sI B sI B sI B I namely,
BRs=RsB=sRs-I B R s R s B s Rs I (7)

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