q. let P sub 3 denote the subspace consisting of all polynomials of degree no greater than 3. Let E: P sub 3 ---> P sub 3 denote the differentiation operator on P sub 3. Write down the standard matrix representation of E?!

- August 13th 2008, 12:37 PMlinearalgebradunceStandard matrix representation of a differentiation operator.
q. let P sub 3 denote the subspace consisting of all polynomials of degree no greater than 3. Let E: P sub 3 ---> P sub 3 denote the differentiation operator on P sub 3. Write down the standard matrix representation of E?!

- August 14th 2008, 01:56 AMflyingsquirrel
- August 14th 2008, 02:27 AMlinearalgebradunce
yes that seems so simple now, thank you. is there a quick way of deciding whether this resultant matrix is diagonalisable or not?! I know the definition of a diagonalisable matrix is that there exists P'AP=D where P'=inverse P and D is a diagonal matrix, etc as this was in the other question I posted. However this time I do not need to prove or show if the matrix is diagonalisable, simply state yes or no.

- August 14th 2008, 05:03 AMflyingsquirrel
is a strictly triangular matrix so it is nilpotent. As nilpotent matrices are not diagonalizable, can't be diagonalizable.

If you're not supposed to know this result, let's show it using contradiction : assume that is diagonalizable. has one eigenvalue which is 0 and whose multiplicity is 4. (since the matrix is triangular and the four diagonal entries are 0) Let be the eigenspace associated to the eigenvalue 0. is the set so . For to be diagonalizable, we need to have . I claim that this is impossible. Can you find out why?