# Help On Eigenspaces

• Jan 1st 2008, 05:57 AM
smoothman
Help On Eigenspaces
Hi there, I'm having a bit of a problem understanding eigenspaces, kernel and span. I've searched the net and wikipedia but there doesn't seem to be any clear examples.

I have an example in a book that says this:

Let,
A =
[ 2 2 2 ]
[ 0 2 2 ]
[ 0 0 2 ]

I can see the characteristic polynomial = $\displaystyle (X - 2)^3$ so 2 is the only eigenvalue.

It then calculates the generalised eigenspaces: $\displaystyle V_t(2)$
$\displaystyle V_1(2) =$
ker [ 0 2 2 ]
.....[ 0 0 2 ]
.....[ 0 0 0 ]

The kernel is calculated by row reducing the matrix:

$\displaystyle V_1(2) =$
ker [ 0 1 0 ] = span [1]
.....[ 0 0 1 ]...........[0]
.....[ 0 0 0 ]...........[0]

$\displaystyle V_2(2) =$
ker [ 0 0 1 ] = span [1] [0]
.....[ 0 0 0 ]...........[0] [1]
.....[ 0 0 0 ]...........[0] [0]

$\displaystyle V_3(2) =$
ker [ 0 0 0 ] = span [1] [0] [0]
.....[ 0 0 0 ]...........[0] [1] [0]
.....[ 0 0 0 ]...........[0] [0] [1]

that is the end of the example.

so now here are my questions:

QUESTION 1
What does it mean by : $\displaystyle V_1(2)$, $\displaystyle V_2(2)$, $\displaystyle V_3(2)$ etc.

QUESTION 2
What exactly is the kernal in this example and how is the span calculated from the kernal matrices....??

QUESTION 3
which part of this whole question/example is the eigenspace?

thankyou very much. if this could be explained then it would clear most of the confusion on this topic.

:)
• Jan 1st 2008, 07:01 PM
An excellent reference for this type of problem is Anton and Rorres' "Elementary Linear Algebra". You will probably be able to find a copy in your university's library or even in your local public library. You are much better off looking in a good textbook than on the net or on wikipedia as it is often hard to find good stuff on the net and wikipedia is not designed as a teaching tool.

Quote:

QUESTION 1
What does it mean by : V_1(2), V_2(2), V_3(2) etc.
$\displaystyle V_t(2)$ seems to be the eigenspace of $\displaystyle A^t$ for the eigenvalue 2

Quote:

QUESTION 2
What exactly is the kernal in this example and how is the span calculated from the kernal matrices....??
The kernel of a matrix A is the set of all x such that Ax = 0

In this example, we are using the fact that the kernel of $\displaystyle A-\lambda I$ is the eigenspace of A for the eigenvalue $\displaystyle \lambda$.
This follows simply from the definition of eigenvalues and eigenvectors.

The span of a set of vectors is the set of all linear combinations of the vectors.

Quote:

QUESTION 3
which part of this whole question/example is the eigenspace?
$\displaystyle V_1(2) = span {(1,0,0)}$

I really do recommend that you get hold of that textbook, or failing that, another elementary linear algebra textbook, because you sound like you are more in need of a thorough introduction to linear algebra than the answers to specific questions that you will get here.