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Finding the eigenvalues of maps of polynomial vector spaces

  • #1

Homework Statement


Let V be the vector space of all real-coefficient polynomials with degree strictly less than five. Find the eigenvalues and their geometric multiplicities for the following maps from V to V:

a) G(f) = xD(f), where f is an element of V and D is the differentiation map.

b) F(f) is obtained by multiplying f by 2+x^3 and crossing out all terms of degree five or greater


Homework Equations


None I can think of.


The Attempt at a Solution


For a), I wrote out that if v is an eigenvector, then for it there exists some h such that G(v) = h*v = h(a4*x^4+...+a1*x+a0) = 4*a4*x^4 + 3*a3*x^3+...+a1*x and from this tried to set up equations such as h*a4 = 4*a4,.., h*a1=a1, h*a0= 0, and so on. But from here I'm not sure how to solve for the eigenvalues. I've never done this before and there has been no explanation either in lecture or in the book of how to approach this kind of problem.

For b), I did the same and came up with F(v) = h*v = (2*a4+a1)*x^4 + (2*a3+a0)*x^3 + 2*a2*x^2 + 2*a1*x + 2*a0 (this is what happens when one simplifies the polynomial and crosses out the summands with terms of degree five or higher). Again, not sure what to do from here.
 

Answers and Replies

  • #2
1,101
3
Given an operator f, I think it's usually easier to consider two cases: 1. the eigenvalue is 0 (considered when Ker f =/= {0}), 2. the eigenvalue is not 0.

For instance, in part a), you can easily check which polynomials are eigenvectors with eigenvalue 0. Then just suppose that the eigenvalue h is not 0, and then using the systems of equations you've written out it's pretty easy to see what h could be. Note the obvious reinterpretation of problem a) as a first order separable ordinary differential equation, which leads to slick solution.
 

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