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Eigenvalue Problem: The Transfer Function

Module by: Steven J. Cox. E-mail the author

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

1sb 1 sb
(1)
This function is a scaled version of the even simpler function 11z 1 1z . This latter function satisfies the identity (just multiply across by 1z1z to check it)
11z=1+z+z2+...+zn1+zn1z 1 1z 1 z z2 ... z n1 zn 1z
(2)
for each positive integer nn. Furthermore, if |z|<1 z 1 then zn0 z n 0 as n n and so Equation 2 becomes, in the limit, 11z=n=0zn 1 1z n 0 zn the familiar geometric series. Returning to Equation 1 we write 1sb=1s1bs=1s+bs2+...+bn1sn+bnsn1sb 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, 1sb=1sn=0bsn 1 sb 1 s n 0 b s n This same line of reasoning may be applied in the matrix case. That is,
sIB-1=s-1IBs-1(1s+Bs2+...+Bn1sn+BnsnsIB-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
sIB-1=s-1n=0Bsn 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 sIB-1ds=n=0Bn Cρ s-1nds=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 IB-1 s1 IB-1= s2 IB-1( s1 IB s2 I+B) s1 IB-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 (sIB)sIB-1=sIB-1(sIB)=I sI B sI B sI B sI B I namely,
BRs=RsB=sRsI B R s R s B s Rs I
(7)

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