Matrix Representation of Nilpotent Linear Operator

[QuantumP7]

Homework Statement

(5.5)$$V = Z_{\xi_{1}} \oplus Z_{\xi_{2}} \oplus ... \oplus Z_{\xi_{k}},$$ the basis for V is:

$$\xi_{1}, \eta(\xi_{1}), ..., \eta(\xi_{1})^{p_{1} - 1}$$
$$\xi_{2}, \eta(\xi_{2}), ..., \eta(\xi_{2})^{p_{2} - 1}$$
.
.
.
$$\xi_{k}, \eta(\xi_{k}), ..., \eta_(\xi_{k})^{p_{k} - 1}$$

relative to which $$\eta$$ is represented by $$Dg[J_{p1}(0), J_{p2}(0), ..., J_{pk}(0)].$$

Theorem 5.9: If $$\eta \in L(V, V)$$ is nilpotent of index $$p_{1}$$, then there exists an integer k, k distinct vectors $$\xi_{1}, ..., \xi_{k}$$ and k integers $$p_{1} \geq p_{2} \geq ... \geq p_{k}$$such that the vectors in (5.5) form a basis for V. Moreover, V is the direct sum of the $$\eta$$-cyclic subspaces generated by the $$\xi_{i}: V = Z_{\xi_{1}} \oplus Z_{\xi_{2}} \oplus ... \oplus Z_{\xi_{k}}.$$ Relative to the basis in (5.5), $$\eta$$ is represented by the matrix $$Dg[J_{p1}(0), J_{p2}(0), ..., J_{pk}].$$

Problem: Find integer k and the vectors $$\xi_{1}, ..., \xi_{k}$$ of Theorem 5.9 for the nilpotent operator $$\eta \in L(\Re_{5 x 1}, \Re_{5 x 1})$$if

$$Mtx_{\epsilon}(\eta) = \left[ \begin{array} {ccccc} 0 & 0 & 0 & 0 & 0\\ 2 & 0 & 0 & 0 & 0\\ 1 & 3 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 4\\ 0 & 0 & 0 & 0 & 0 \end{array} \right]$$

Homework Equations

The Attempt at a Solution

The answer is supposed to be k = 2 and $$\xi_{1} = \epsilon_{1}$$ and $$\xi_{2} = \epsilon_{5},$$ with the $$\epsilon_{i}$$ being the standard basis. I don't even know how they arrive at this answer. There is no example in this book to help guide how to do this problem. I'm self-studying, so I'm not asking for anyone to do this work for me. I just really want to understand how to do this problem. Anyone know where to start on this?

 

[mighty2000]

Hello, thank you for your question. This is an interesting problem that involves the concept of nilpotent operators and their cyclic subspaces. Let's break down the problem and try to understand it step by step.

First, let's define some of the terms used in the problem:

- Nilpotent operator: A linear operator \eta is called nilpotent if there exists a positive integer p such that \eta^{p} = 0, where \eta^{p} denotes the p-th power of \eta.
- Index of nilpotency: The smallest positive integer p for which \eta^{p} = 0 is called the index of nilpotency of \eta.
- Cyclic subspace: A subspace V \subset W is called cyclic with respect to a linear operator \eta if there exists a vector \xi \in V such that V = \text{span}\{\xi, \eta(\xi), \eta^{2}(\xi), ... \}.
- Direct sum: The direct sum of two subspaces V and W, denoted by V \oplus W, is the subspace containing all vectors of the form v + w, where v \in V and w \in W.

Now, let's look at the theorem given in the problem statement. It states that for a nilpotent operator \eta with index of nilpotency p_{1}, there exists an integer k, k distinct vectors \xi_{1}, ..., \xi_{k}, and k integers p_{1} \geq p_{2} \geq ... \geq p_{k} such that the vectors in (5.5) form a basis for V. Moreover, V is the direct sum of the \eta-cyclic subspaces generated by the \xi_{i}.

In simpler terms, this means that for a nilpotent operator \eta, there exists a basis for its vector space V consisting of k distinct vectors \xi_{1}, ..., \xi_{k}, where each vector \xi_{i} generates a cyclic subspace of V of index p_{i}.

Now, let's apply this theorem to the given problem. We are given a nilpotent operator \eta \in L(\Re_{5 x 1}, \Re_{5 x 1}) and its matrix representation Mtx_{\epsilon}(\eta). We need to find the integer k and the vectors