[daretodream-x]

I have a matrix A =
And I need to find the Jordan Normal Matrix and the transformation matrix, such that
I've worked out the characteristic polynomial to be
And my eigenspace for eigen value -1, to span:
which I have named v1 and v2.

When I try to find v3 from either (A+1I)v3=v1 or (A+1I)v3=v2, I end up with a contradiction, something like 0=3.. etc.

The answers state that P= but that doesn't even include my v2..?

I've had the same problem with an identical question today.. can someone please tell me what's going wrong?

[Jake22]

(Original post by daretodream-x)
When I try to find v3 from either (A+1I)v3=v1 or (A+1I)v3=v2, I end up with a contradiction, something like 0=3.. etc.

Revise your notes. You have misunderstood the process.

What you want to do here is to complete to a basis (i.e. choose to be any vector not in the span of ) and then define . It then follows that is the basis you require.

[daretodream-x]

(Original post by Jake22)
Revise your notes. You have misunderstood the process.

What you want to do here is to complete to a basis (i.e. choose to be any vector not in the span of ) and then define . It then follows that is the basis you require.

Sorry, I'm not following..?

[Jake22]

(Original post by daretodream-x)
Sorry, I'm not following..?

Your eigenspace is a 2-dimensional subspace of your 3-dimensional space. You chose a basis . It follows that any other vector that is not in that two dimensional subspace (any vector linearly independent from both of them) completes to a basis for the whole of your space i.e. is such that is a basis of your 3-dimensional space.

Once you have done that - you just define a new vector by setting and then should be a basis for which your matrix is in Jordan Normal Form.

[Jake22]

Incidentally, I think the -1 in the top row of the matrix should be a 1.

Then, if you follow the method I told you - they pick and set . The required (ordered) basis is then .