# Eigenvalue of Polynomial Transformation

1. Mar 29, 2012

### PandaGunship

1. The problem statement, all variables and given/known data
Let T:P2→P2 be defined by
T(a0+a1x+a2x2)=(2a0-a1+3a2)+(4a0-5a1)x + (a1+2a2)x2
1) Find the eigenvalues of T
2) Find the bases for the eigenspaces of T.

I believe the 'a' values are constants.
2. Relevant equations
None.

3. The attempt at a solution
The problem I am having is actually pulling out the matrix for T. I know how to find eigenvalues (by solving det(λI-A) - A being the matrix) and from that finding the bases of the eigenspaces comes from substituting the eigenvalues into (λI-A) and performing elementary row operations to find the eigenvectors which form the bases.

What I have tried to do is separate the basis (1,x,x2) from the rest and come up with the matrix :
2a0 -a1 3a2
4a0 -5a1 0a2
0a0 a1 2a2
Am I on the right track here or am I barking up the wrong tree? If so what method should i follow?

2. Mar 29, 2012

### Robert1986

Well, sort of. You need to figure out which elements the polynomials $1,x,x^2$ get sent to by $T$. Once you do that, put them into a matrix and proceed as you have described. So, if you put the polynomial $p_1(x) = x$ into $T$ what would come out "the other side"?

3. Mar 29, 2012

### tiny-tim

Hi PandaGunship! Welcome to PF!
leave all the a's out!

4. Mar 29, 2012

### PandaGunship

Just to check, my matrix is correct?
As per your direction, a0 goes to 2a0 + 4a0x +0x2 which I make into a column like the matrix I put it already?
Thanks Tim :) I thought of leaving the a's out, but what of their importance? Are they just to show what each constant of the basis is mapped into?

5. Mar 29, 2012

### tiny-tim

No, your vectors will have a's.

(The 1 x x2 are just a basis like x y z …

you wouldn't expect x y z to appear in a vector or matrix, would you? )​

6. Mar 29, 2012

### Robert1986

Well, I said to find out where $p_1(x) = x$ goes, and what you did was find where $p(x) = a_0$ goes. I only mentioned to do it as an example, so its OK that you did it for $p(x) = a_0$.

So, the $a_0, a_1, a_2$ are there because you need to know what $T$ does to a general quadratic polynomial, and that is the form of a general quadratic polynomial.

Now, the three basis polynomials we are using are: $p_0(x) = 1, p_1(x) = x, p_2(x) = x^2$. Now, for each of these basis polynomials, what is $a_0, a_1, a_2$? As an example, for $p_0$ we have that $a_0 = 1, a_1 = 0, a_2 = 0$. So, $T(p_0(x)) = 2a_0 + 4a_0x = a + 4x$. So, you got it correct, but as Tiny Tim said, you need to get rid of the a's because for the basis polynomials they are all 1 or 0.

7. Mar 29, 2012

### PandaGunship

I suppose not :P Thanks :)
Ok I understand properly now. In the beginning when I was going over this problem I thought that I must remove the a's but I didnt want to remove them 'just because'. I wanted to know that what I was doing was actually correct, even if my reasoning was a bit off the mark.
Thank you both for your help