1. ## 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 = $(X - 2)^3$ so 2 is the only eigenvalue.

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

The kernel is calculated by row reducing the matrix:

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

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

$V_3(2) =$
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 : $V_1(2)$, $V_2(2)$, $V_3(2)$ 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.

2. 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.

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

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 $A-\lambda I$ is the eigenspace of A for the eigenvalue $\lambda$.
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.

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

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.