1. ## Generalized invariant subspace

Hi,
Let A be an nxn non-scalar matrix (i.e A is not a scalar multiple of the identity matrix).Show that there exists a non-trinial subspace W of R^n ,

that is invariant under every nxn matrix B such that AB=BA.

https://en.wikipedia.org/wiki/Invariant_subspace

2. ## Re: Generalized invariant subspace

Hey hedi.

This should preserve information in a way that is one to one.

Hint - Consider the determinant and how that helps figure out whether the transformation is one to one.

3. ## Re: Generalized invariant subspace

I dont see the connection to "one to one"

4. ## Re: Generalized invariant subspace

If v e W and T(v) e W then the information is preserved and the mapping has to be one to one.

Since it goes from R^n to R^n you have a square matrix and because of that it has to have a non-zero determinant.

5. ## Re: Generalized invariant subspace

It has been a very long time that I have done this problem. I seem to recall showing that there must exist some positive integer $K$ such that for all $k\ge K$, we have $\text{dim}\left(A^k\mathbb{R}^n\right)=\text{dim} \left(A^K\mathbb{R}^n\right)$

I completely forget how we used that to prove the theorem, though. I'll take another look at it after work.