Hi,
let X be a vector matrix,
then you can proove that {ABX, X}={AX, X}, what could help you
Hello all,
I am trying to find a proof that, for B an invertible matrix, rank(AB) = rank(BA) = rank(A).
I know I should probably play with the null space of A, AB and BA but I cannot find the right approach to this problem.
Do you have any hint for me, or online notes with the property proven?
If A is from to then its rank is n- nullity(A) so, yes, looking at the null space is one way to go. If you can show that AB, BA and A all have the same nullity then they will all have the same rank.
In particular, if u is in the null space of A, the Au= 0 so, of course, BAu= B0= 0 for any matrix B. That is, the null space of A is always a subspace of the null space of BA. Now, the other way. Suppose BAu= 0. The B(Au)= 0 and since B is invertible, Au= 0 so we have that the null space of BA is a subspace of the null space of A. That proves that A and BA have exactly the same null spaces so of course they are of the same dimension.
Now, suppose u is in the null space of AB. That is, suppose ABu= A(Bu)= 0. What can you say about u?
The rank of a matrix is merely the dimension of the range of , , which is the subspace of consisting of all vectors of the form with .I am trying to find a proof that, for B an invertible matrix, rank(AB) = rank(BA) = rank(A).
You could therefore prove that and have the same dimension as if is invertible.
Here are some hints:
1. Show that .
First off, if then for some and so . Thus .
Now prove that in similar fashion (needs to be invertible here).
2. It is unfortunate that is not the same as in general, so a similar proof to the above fails in this case. However there is another result that saves the day.
Recall the null space of , , is the set of all vectors in for which .
Well, it turns out that if is invertible (prove it!)
Now there is a very well-known theorem which states that the dimensions of and add up to the dimension of .
The same is true of and of course.
Now if and are the same then they have the same dimension. It follows that ...
Hold on, someone's at the door. I'll just go answer it. Back in a mo.
(Dum dee doo, my eyes are old and bent, etc ...)
Hello, can I help you? ... I said, what d'you want? Wait ... What ... No, don't ... HEEEEEEELP ...