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# Jordan Basis watch

1. I have no idea, anyone would help me?
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2. This could be hard to explain without pictures.

Basically, you want to be thinking about D as a matrix. Its input is the vector:

(a0, a1, a2, ..., a9999, a10000)

and it returns the vector:

(2a2, 6a3, 12a4, ..., 99990000a10000, 0, 0)

From this, you can work out what the matrix for D looks like. It has zeroes all along the diagonal, and zeroes on the diagonal above that. The diagonal above that reads (2, 6, 12, 20, 30, 42, ..., 99990000)

Which is the series {2.1, 3.2, 4.3, 5.4, 6.5, ..., 10000.9999}

To calculate the characteristic polynomial chD, you need to subtract λ times the identity matrix from this, and work out the determinant. If you choose to expand it about the first row, then a little scribbling will probably convince you that you get chD=(-λ)10001.

The minimum polynomial should be easy to work out now - check your notes!

To put it in Jordan normal form (block diagonal form) you just need to faff about with row operations for a bit, I think. It's been a while since I did linear algebra. I can't remember what a Jordan basis is, sorry!
3. (Original post by Cexy)
This could be hard to explain without pictures.

Basically, you want to be thinking about D as a matrix. Its input is the vector:

(a0, a1, a2, ..., a9999, a10000)

and it returns the vector:

(2a2, 6a3, 12a4, ..., 99990000a10000, 0, 0)

From this, you can work out what the matrix for D looks like. It has zeroes all along the diagonal, and zeroes on the diagonal above that. The diagonal above that reads (2, 6, 12, 20, 30, 42, ..., 99990000)

Which is the series {2.1, 3.2, 4.3, 5.4, 6.5, ..., 10000.9999}

To calculate the characteristic polynomial chD, you need to subtract λ times the identity matrix from this, and work out the determinant. If you choose to expand it about the first row, then a little scribbling will probably convince you that you get chD=(-λ)10001.

The minimum polynomial should be easy to work out now - check your notes!

To put it in Jordan normal form (block diagonal form) you just need to faff about with row operations for a bit, I think. It's been a while since I did linear algebra. I can't remember what a Jordan basis is, sorry!
chD=(-λ)10001

but the question wants us to find chD=(X)10001
4. Define X = -λ then

I wouldn't worry about being out by a minus sign - it comes down to exactly the same thing, since you're going to be solving chD=0 anyway.
5. (Original post by Cexy)
Define X = -λ then

I wouldn't worry about being out by a minus sign - it comes down to exactly the same thing, since you're going to be solving chD=0 anyway.
so the minimal polynomial should be x^10001, is it right?
6. Let -λ = X then
7. anyone knows how to solve Jordan basis?

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Updated: November 15, 2005
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