Finding eigenvalues and eigenvectors for a polynomial transformation

[trap101]

Hi,

So for some reason I have the hardest time trying to work with polynomials in linear algebra. I can't explain it, but whenever I see a question I draw a complete blank.

Question: i) Find all the eigenvalues. ii) for each eigenvalue λ, find a basis of the eigenspace E_λ.

T: P₃(R) --> P₃(R) defined by T(p)(x) = p'(x) + 2p(x)

So this is all I'm given. My question is what polynomials do I use to find the eigenvalues, and once I find those eigenvalues how do I find the eigenvectors? I'm inclined to try and solve it like eigenvector problems with matrices, problem is I don't know how to put this into a matrix.

 

[Ray Vickson]

Hi,

So for some reason I have the hardest time trying to work with polynomials in linear algebra. I can't explain it, but whenever I see a question I draw a complete blank.

Question: i) Find all the eigenvalues. ii) for each eigenvalue λ, find a basis of the eigenspace E_λ.

T: P₃(R) --> P₃(R) defined by T(p)(x) = p'(x) + 2p(x)

So this is all I'm given. My question is what polynomials do I use to find the eigenvalues, and once I find those eigenvalues how do I find the eigenvectors? I'm inclined to try and solve it like eigenvector problems with matrices, problem is I don't know how to put this into a matrix.

You need to solve the problem $p'(x) + 2p(x) = \lambda p(x),$ and the solution must be a polynomial.

RGV

 

[trap101]

You need to solve the problem $p'(x) + 2p(x) = \lambda p(x),$ and the solution must be a polynomial.

RGV

But what am I solving for? it can't be for lambda. In simple algebra I would have the polynomial and solve for "x". Do I use the standard basis vectors?

 

[Ray Vickson]

But what am I solving for? it can't be for lambda. In simple algebra I would have the polynomial and solve for "x". Do I use the standard basis vectors?

In this case you are solving a first-order linear differential equation that happens to have a parameter, λ, in it. In other words, you need to find the function p(x).

RGV

 

[trap101]

Oh, I should mention that I haven't done differential equations yet at my Uni. Only have Calc courses and Linear Algebra part I.

What I wanted to do was solve it as if it was searching for an eigenvector but with matrices. But an issue I always have is I don't know how to transform the polynomial into a matrix, expecially when I'm not given a specific polynomial.

 

[kai_sikorski]

problem is I don't know how to put this into a matrix.

What is the basis for your space? HINT: you need to find 4 polynomials, such that any cubic polynomial can be expressed as their sum. There are many possible choices, but probably only a few natural ones.

Okay assuming you can answer that question here is how you would find a matrix representation for the operator. Say the basis is $e_i$ for $i = 1 ... 4$.

Define the vectors $v_j$ as

$$v_j = T (e_j)$$

So above line means you take your polynomials that you chose as your basis, and you apply the operator T to them which amounts to adding twice the polynomial to its own derivative.

Now since $v_j$ must be cubic polynomials, you must be able to express them uniquely in the basis you chose. Say this representation is
$$v_j = b_{1j}e_1 + b_{2j}e_2 + b_{3j}e_3 + b_{4j}e_4$$
Your matrix is now given by $T_{ij} = b_{ij}$

 

[trap101]

Ahhh. Thank you. So to build on that question, because it always seems to pop up in some fashion or another. If I'm not given any vectors that I have to specifically apply my linear transformation to, should I just assume that I can use the standard basis vectors? Because my troubles always occur when I don't know what vectors should be chosen.

The reason I ask is because there is a similar question where i have to verify that the given vector is an eigenvector:

p = x³ ...under the same P₃ conditions, but defined by: T(x) =
xp' - 4p. Now the eigenvector condition is T(x) = λx. So do I use the standard basis vectors of P₃ in that transformation and try to obtain the λ that would prove this?

 

[kai_sikorski]

It shouldn't matter what intermediate basis you work in. When you re-express the eigen-vectors as polynomials they should be the same no matter what basis you chose. If an eigen-space for a particular eigen-value had dimension greater than 1, then the individual polynomials you get might be different but collectively they should span the same space.

 

[trap101]

Thanks. I'm going to give this all a try, hopefully there won't be any problems