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Jordan Basis for Differential Operator

by fishshoe
Tags: jordan basis, linear algebra
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fishshoe
#1
Mar8-12, 11:43 AM
P: 16
1. The problem statement, all variables and given/known data
Let [itex] V = P_n(\textbf{F}) [/itex]. Prove the differential operator D is nilpotent and find a Jordan basis.

2. Relevant equations
[itex] D(Ʃ a_k x^k ) = Ʃ k* a_k * x^{k-1} [/itex]

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
[itex]
\left[
\begin{array}{ c c }
0 & 1 & 0 \\
0 & 0 & 2 \\
0 & 0 & 0
\end{array} \right]
[/itex]
Is that the kind of basis they're looking for here?
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morphism
#2
Mar8-12, 11:30 PM
Sci Advisor
HW Helper
P: 2,020
No - in a Jordan basis, all entries in the superdiagonal (i.e. the line above the diagonal) have to be either 1 or zero.
fishshoe
#3
Mar10-12, 09:55 AM
P: 16
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?


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