- #1
Kate2010
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Homework Statement
Let V be the space of polynomials with degree [tex]\leq[/tex] n (dimV=n+1)
i. Let D:V->V be differentiation, i.e. D: f(x) -> f'(x)
What are the eigenvalues of D? Is D diagonalisable?
ii. Let T be the endomorphism T:f(x) -> (1-x)2 f''(x).
What are the eigenvalues of T? Is T diagonalisable?
Homework Equations
The Attempt at a Solution
i. I have constructed the matrix of D with respect to the basis {1,x,x2,...,x2}
C = [tex]\left[ \begin{array}{ccccc} 0 & 1 & 0 & ... & 0 \\ 0 & 0 & 2 & ... & 0 \\ : & : & : & : & : \\ 0 & 0 & 0 & 0 & n \\ 0 & 0 & 0 & 0 & 0 \end{array} \right][/tex]
The characteristic polynomial of this (I think) is xn+1 + (-1)n(n!)
I have no idea how to solve this to get eigenvalues?
Or is there a better way to approach this question?