Thread: Linear 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

Originally Posted by taypez

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

Can you tell us in words what space is represented by L(F ∞)?

I could guess, but would rather know.

RonL

Originally Posted by CaptainBlank

Can you tell us in words what space is represented by L(F ∞)?

I could guess, but would rather know.

RonL

Maybe all functions (not necesarrily continous) but defined on (-∞,∞)

Originally Posted by ThePerfectHacker

Maybe all functions (not necesarrily continous) but defined on (-∞,∞)

The L would normaly denote a space of linear operators, but I would
expected a small "L" for one with a countable basis as I suspect this has.

RonL

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

Originally Posted by taypez

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

Still not quite clear, my guess that we want L(F) here to be the space of
linear operators on the space of all sequences on either R or C that are bounded?

So if

Originally Posted by CaptainBlack

Still not quite clear, my guess that we want L(F) here to be the space of
linear operators on the space of all sequences on either R or C that are bounded?

So if

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