Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $T$ be a linear operator in the space of polynomials of degree less or equal to n, $T(p(x))=xp´(x)$. I need to calculate the eigenvalues of this operator.

So far, I calculated the matrix of the transformation in the canonical basis: $$ \begin{matrix} 0 & 0 & 0 & ... & 0 \\ 0 & 1 & 0 & ... & 0 \\ 0 & 0 & 2 &... &0 \\ & & & ... & \\ 0 & 0 & 0 &... &n \\ \end{matrix} $$

And if I calculate the characteristic polynomial of this matrix, then $0,1,2,3,...,n$ are all eigenvalues. I don´t know how to calculate the associated eigenspace for EACH of these values. Am I missing something? Thak you!

share|improve this question
    
So, just to recap: you found a basis composed of eigenvectors of $T$, correct? –  Jonathan Y. Dec 9 '13 at 15:50
    
Not quite. I don´t know the English word of it. In Spanish we call it canonical basis, is the basis of the space of polinomials ${1,x,x^2,...,x^n}$ It think that those are in fact eigenvectors, but I was not thinking of that... –  Luna Sage Dec 9 '13 at 15:54
    
Standard basis, I think. Regardless, you've got it! (and the way you could've seen it is the representing matrix was diagonal.) –  Jonathan Y. Dec 9 '13 at 16:51
    
That's right. Thanks! –  Luna Sage Dec 9 '13 at 17:16

1 Answer 1

For that matrix, an eigenvector for the $i$'th diagonal eigenvalue is $E_{i}$, where this is the vector with the $i$'th entry $1$ and all other entries $0$.

It looks like in your case all eigenvalues are distinct, and hence each eigenspace is $1$ dimensional and generated by each eigenvector we found.

share|improve this answer
    
Thanks! I was thinking of that, I just didn't know how to write it down. –  Luna Sage Dec 9 '13 at 16:00

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.