I was going through Jordanization by Jonathan Nilsson.

Question

I was going through Jordanization by Jonathan Nilsson. Here he describes the algorithm for Jordanizing any square complex matrix $A$.

Here $T := (A - \lambda I )$. Now finding the sub spaces $Im(A - \lambda I )^k$ for each $1 \leq k \leq m_{\lambda}$, where $m_{\lambda}$ is the exponent of $(t − λ)$ in $p_A$, the characteristic polynomial of $A$.

But here he claims that ...for higher $k$ the images will all be the same.

I am getting stuck here. Help Needed..

In the paper he also have explained the matrix algorithm using an example.

Thank You...

Answer 1

The statement is true even in a more general context. Let T be any linear map on a finite dimensional vector space V.

Then $V \supset T(V) \supset T^2(V) \supset T^3(V) \supset \cdots$

Taking the dimension of this sequence we obtain a decreasing sequence of nonnegative integers, which has a lowest value. This means that the sequence of subspaces stabilizes:

$T^{n+1}(V)=T^n(V)$ for some n. In other words, the subspaces $Im T^k$ are all equal for $k \geq n$.