Need some help.

Find all eigenvalues and eigenvectors of the backward shift operator T Є L(F ∞) defined by:

T(z1,z2,z3....) = (z2,z3....).

Thanks

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- Mar 15th 2007, 10:39 AMtaypezLinear Algebra II
Need some help.

Find all eigenvalues and eigenvectors of the backward shift operator T Є L(F ∞) defined by:

T(z1,z2,z3....) = (z2,z3....).

Thanks - Mar 15th 2007, 12:39 PMCaptainBlack
- Mar 15th 2007, 12:44 PMThePerfectHacker
- Mar 15th 2007, 12:47 PMCaptainBlack
- Mar 15th 2007, 04:09 PMtaypez
Sorry. When I cut and pasted this into the post, it changed the script L.

L(**F**) is the set of all linear maps on**F**infinity.**F**denotes R or C.

Thanks - Mar 15th 2007, 10:38 PMCaptainBlack
- Mar 15th 2007, 11:32 PMCaptainBlack
If this is the case then T is the backwards shift operator which for all

Z=(z(1), z(2), ..) in F with max(z(1), z(2), ..) = |Z| < infty:

T(Z) = Z' = (z(2), z(3), .. )

Now if lambda and X are eigen value and eigen vector for T, then:

T(X) = lambda X,

so x(n+1) = lambda x(n), for all n>1.

Hence x(n) = lambda^(n-1) x(1), which is in F iff |lambda| <= 1.

Thus every lambda such that |lambda|<=1 is an eigen value of T and

(1, lambda, lambda^2, .., lambda^n, ..) is an eigen vector corresponding

to the eigen value lambda.

RonL