Generalised eigenvector

Description

The Eigenvektors of a matrix only form a Basis (Lineare Algebra) of the vector space if the matrix is diagonalisable. The generalised eigenvector allows us to find a natural basis even if the matrix is not diagonalisable.

To each generalised eigenvector we can associate a level.

Definition

Let $A\in GL(n,\mathbb{R})$ be a matrix with the corresponding linear map $A:{\mathbb{R}}^n\to {\mathbb{R}}^n,v\mapsto Av$. A vector $v\in {\mathbb{R}}^n$ is called a generalised eigenvector of level $k$ with eigenvalue $\lambda$ if: $(A-\lambda I{)}^k(v)=0\text{ and }(A-\lambda I{)}^{k-1}(v)\ne 0$

The idea

To understand the generalised eugenvector we try to understand the Jordan decomposition. A Jordan block as a matrix is a sum of a rescaling and a nilpotent shift. $\left(\begin{array}{ccc}\lambda & 1& \\ & \lambda & 1\\ & & \lambda \end{array}\right)=\left(\begin{array}{ccc}\lambda & & \\ & \lambda & \\ & & \lambda \end{array}\right)+\left(\begin{array}{ccc}& 1& \\ & & 1\\ & & \end{array}\right)$ The nilpotent shift has the property that $(x_1,x_2,x_3)\mapsto (x_2,x_3,0)$. In the case of the normal eigenvector, this nilpotent summand is just zero (this is where the generalisation lies). Notice, that there is a hierarchy in the axis. The nilpotent shift first deletes the last axis $⟨e_3⟩$. Then it deletes the second and then the first. The equation $(A-\lambda I)$ returns the nilpotent summand and $(A-\lambda I{)}^k$ applies $k$ iterations it.

The matrix studies just now was quite simple. Can we apply this idea to other matrices? Well, since every matrix can be made into a Jordan-form by change of basis, the procedure is the same:

• For each generalised eigenspace there are generalised eigenvectors
• The linear map approximately scales those eigenvectors.
• If we remove the scaling, we discover that the generalised eigenvectors follows a hierarchy: Generalised eigenvectors are sent to one another until they completely die out

In a sense, the nilpotent summand of a matrix feeld like the skeleton that describes some motion. This motion must be the skew transformations in the matrix.

When trying to find a basis of generalised eigenvectors, we try to find the highest one of highest level and apply the nilpotent element to map it to the others.

Properties

Tip

Examples

Example