Jordan Basis for Differential Operator

In summary, the conversation discusses proving the differential operator D is nilpotent and finding a Jordan basis for V = P_n(\textbf{F}). The individual has already completed the proof for D being nilpotent, but is unsure of what a Jordan basis is and how to find it. They mention a previous problem with a matrix form for the differential operator, but it is not the type of basis requested. The conversation ends with a suggestion for a potential Jordan basis for polynomials of order up to n.
  • #1
fishshoe
16
0

Homework Statement


Let [itex] V = P_n(\textbf{F}) [/itex]. Prove the differential operator D is nilpotent and find a Jordan basis.

Homework Equations


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

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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  • #2
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
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?
 

FAQ: Jordan Basis for Differential Operator

1. What is the Jordan Basis for Differential Operator?

The Jordan Basis for Differential Operator is a set of linearly independent solutions to a homogeneous linear differential equation. It is used to find the general solution to a differential equation by using a linear combination of the basis solutions.

2. How is the Jordan Basis for Differential Operator related to eigenvalues and eigenvectors?

The Jordan Basis for Differential Operator is related to eigenvalues and eigenvectors through the characteristic polynomial of the differential operator. The eigenvalues of the operator correspond to the roots of the characteristic polynomial, and the eigenvectors correspond to the basis solutions of the operator.

3. Can the Jordan Basis for Differential Operator be used for non-homogeneous differential equations?

No, the Jordan Basis for Differential Operator can only be used for homogeneous linear differential equations. For non-homogeneous equations, other methods such as variation of parameters or the method of undetermined coefficients must be used.

4. How is the Jordan Basis for Differential Operator determined?

The Jordan Basis for Differential Operator is determined by finding the eigenvalues and corresponding eigenvectors of the differential operator. These eigenvectors form a basis for the solution space of the operator, and can be used to find the general solution to the differential equation.

5. Are there any limitations to using the Jordan Basis for Differential Operator?

One limitation of using the Jordan Basis for Differential Operator is that it can only be used for constant coefficient differential equations. Additionally, it may be difficult to find the eigenvalues and corresponding eigenvectors for more complex operators, making it less practical for certain types of differential equations.

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