Got completely confused with this nilpotent and JCF stuff, need some help.
Matrix $A_{n\times n}$ is nilpotent of order K, $1\le k\le 4$
Need to find:
a list of all possible dimensions of null(A), $null(A^2)$, ... , $null(A^{k-1})$
How do I do that?
Thank you!
Note: I kinda figured out how to do this below, but I am still a bit shaky on that one, so please correct
On
A solution for n = 4:
The dimensions of null(A), null(A2), ... , null(Ak−1) depend on the given k for a certain case.
Important points are:
- dim null($A^{k-1}$) $\gt$ ... $\gt$ dim null($A^2$) $\gt$ dim null($A$)
- dim null($A^{k}$) = n
So, there are 4 possible values for k:
if k = 1, the only possible case is
dim null(A) = 4
if k = 2, two cases are possible
- dim null ($A$) = 3
dim null ($A^2$) = 4- dim null ($A$) = 2
dim null ($A^2$) = 4
if k = 3:
dim null($A$) = 2
dim null($A^2$) = 3
dim null($A^3$) = 4
and if k =4:
dim null($A$) = 1
dim null($A^2$) = 2
dim null($A^3$) = 3
dim null($A^4$) = 4
Let $\alpha_i=dim(\ker(A^i))$. In particular $(*)$ $\alpha_0=0,\alpha_k=n$ and , for every $i$, $\alpha_{i+1}-\alpha_i\geq 0$.
tmac_balla gave the (correct) solutions when $n=4$. Unfortunately, he killed the most important conditions, that are,
$(**)$ for every $i$, $\alpha_{i+1}-\alpha_i\leq \alpha_{i}-\alpha_{i-1}$.
Let $C(A)$ be the commutant of $A$. The previous values are important, because $dim(C(A))=\sum_{i\geq 0} (\alpha_{i+1}-\alpha_i)^2$.
Finally, for every sequence $(\alpha_i)_i$ satisfying $(*)$ and $(**)$, there is instance of $A$ s.t. its associated sequence is $(\alpha_i)_i$.