Let be defined by T(p)=xp'. Find eigenvalues of T.

I'm not sure what the notation of T(p)=xp' means and how to go about finding the eigenvalues.

Thanks.

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- May 17th 2010, 08:11 AMwattkowFinding eigenvalues
Let be defined by T(p)=xp'. Find eigenvalues of T.

I'm not sure what the notation of T(p)=xp' means and how to go about finding the eigenvalues.

Thanks. - May 17th 2010, 11:58 AMdwsmith
- May 17th 2010, 04:24 PMwattkow
- May 17th 2010, 04:29 PMdwsmith
I am assuming P is a polynomial since we are mapping on . Unless your book notates this differently, I am almost positive these are polynomials; however, different books notate differently. For instance, mine says are polynomial of degree 1 or less. Thus, the 2 in is the amount of elements {1, x} but some books refer to as degree two or less, { , , }. What does your book say?

- May 17th 2010, 04:35 PMwattkow
- May 17th 2010, 04:38 PMdwsmith
- May 17th 2010, 05:38 PMRandom Variable
I would do the following:

A basis for is

and

so the transformation matrix is

which has eigenvalues 0,1, and 2 - May 18th 2010, 06:22 AMHallsofIvy
The "set of all 2nd degree polynomials" does not form a vector space. As dwsmith said, is the set of all polynomials of degree 2

**or less**.

Random Variable showed how to represent this transformation as a matrix- worth knowing for itself.

But for this problem, you don't have to do that. As dwsmith said, if , then . If is an eigenvalue for T, we must have some a, b, c, not all 0, such that . Since that must be true for all x, we can set corresponding coefficients equal: , , and .

If a is not 0, then , if b is not 0, then , and if c is not 0, then .

That is, 0 is an eigenvalue with eigenvectors any multiple of 1, 1 is an eigenvalue with eigenvectors any multiple of x, and 2 is an eigenvalue with eigenvectors any multiple of , exactly what Random Variable got. - May 18th 2010, 11:09 PMwattkow
Thanks fellas.