Proving Matrix in Block Form: nxn Nilpotent & (n-k)x(n-k) Invertible

[Alupsaiu]

Homework Statement

Prove any nxn matrix can be written as in block form

N 0
0 B

where N is a kxk nilpotent matrix and B is an (n-k)x(n-k) invertible matrix.




Need help getting started, or any hints/any help at all would be really appreciated. Thank you!

 

[grey_earl]

What do you mean by "can be written"? Is similar to (as in A = S^(-1) B S)? If yes, Jordan normal form is your answer, simply order all Jordan blocks of eigenvalue zero in the upper left corner.

 

[Alupsaiu]

The wording is from a problem I found online, but I think similarity is what they're after. I'm not sure what else it could mean. Thanks for the help. Now time to google Jordan normal form.

 

[grey_earl]

If you don't know what the Jordan normal form of a matrix is, here are some lecture notes: http://www.math.tamu.edu/~dallen/m640_03c/lectures/chapter8.pdf

However, the proofs are quite involved, so it's not something for one afternoon. Maybe there is a simpler way to prove your statement, I don't know. But once you have Jordan normal form, almost everything related to that becomes trivial :)

 

[Alupsaiu]

I found a fact on wikipedia saying that any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. How would you go about showing this?

 

[grey_earl]

The extension basically is needed because not all polynomials factor completely over any field, for example x²+1 is not factorizable over the reals, but over the complex numbers you have x²+1 = (x-i)(x+i). Since the eigenvalues of any matrix are the roots of the characteristic polynomial, and in Jordan normal form the eigenvalues of your matrix are along the main diagonal, you need to include this.

To show that every matrix then has a Jordan normal form, you basically construct the normal form itself, which goes by induction (the section "A proof" in the wikipedia article gives a short overview).

However, for your original problem, you even don't need to construct the whole Jordan normal form. You only need to construct the Jordan block corresponding to λ=0, and leave the rest at it is (this rest gives you B). Then you don't even need to factor the characteristic polynomial corresponding to B, and so don't need to extend your field.