# Jordan Basis for Differential Operator

1. Mar 8, 2012

### fishshoe

1. The problem statement, all variables and given/known data
Let $V = P_n(\textbf{F})$. Prove the differential operator D is nilpotent and find a Jordan basis.

2. Relevant equations
$D(Ʃ a_k x^k ) = Ʃ k* a_k * x^{k-1}$

3. The attempt at a solution
I already did the proof of D being nilpotent, which was easy. But we haven't covered what a "Jordan basis" is in class and it's not in either of my textbooks. I know what Jordan Canonical Form is, and Jordan blocks, but I don't know what a Jordan basis is.

Earlier I did a problem that showed that the matrix form of the differential operator on polynomials of order 2 or less. It was
$\left[ \begin{array}{ c c } 0 & 1 & 0 \\ 0 & 0 & 2 \\ 0 & 0 & 0 \end{array} \right]$
Is that the kind of basis they're looking for here?

2. Mar 8, 2012

### morphism

No - in a Jordan basis, all entries in the superdiagonal (i.e. the line above the diagonal) have to be either 1 or zero.

3. Mar 10, 2012

### fishshoe

So do I need something like

\begin{array}{ccc}
0 & 1 & 0 & \dots & 0 \\
0 & 0 & 1 & \dots & 0 \\
\dots \\
0 & 0 & 0 & \dots & 1 \\
0 & 0 & 0 & \dots & 0 \end{array}

as an n-vector Jordan basis for the polynomials of order up to n?