# Linear Algebra II

• Mar 15th 2007, 11:39 AM
taypez
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
• Mar 15th 2007, 01:39 PM
CaptainBlack
Quote:

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
• Mar 15th 2007, 01:44 PM
ThePerfectHacker
Quote:

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 (-∞,∞)
• Mar 15th 2007, 01:47 PM
CaptainBlack
Quote:

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
• Mar 15th 2007, 05:09 PM
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
• Mar 15th 2007, 11:38 PM
CaptainBlack
Quote:

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
• Mar 16th 2007, 12:32 AM
CaptainBlack
Quote:

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