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.