1. ## rank(AHH^-1) <= rank(AH)

I need to prove that

rank(AHH^-1) <= rank(AH)

When H is nonsingular nxn, and A is mxn.

2. Originally Posted by be more
I need to prove that

rank(AHH^-1) <= rank(AH)

When H is nonsingular nxn, and A is mxn.
rank (AHH^-1) = rank A
rank of (AH) <= min[rank A, rank H]
as H is invertible rank H = n
so rank of (AH) <= rank A

I got a result that is completely opposite to what you have.

3. Originally Posted by aman_cc
rank (AHH^-1) = rank A
rank of (AH) <= min[rank A, rank H]
as H is invertible rank H = n
so rank of (AH) <= rank A

I got a result that is completely opposite to what you have.
Well it's a two part question, with the idea being that in the end we're going to prove rank(AH) = rank(A)

Part 1 is what you did above. To prove that rank(AH)<=rank(A).
Part 2 is to prove rank (AHH^-1) <= rank (AH).

By proving the two statements, we'd show that rank(AH)=rank(A).
A hint that was given was to consider how we reduce to row echelon form.

4. Well guess this might work then then Put B = AH, and X= H^-1 in part 1

Then
rank(BX) <= rank (B) (like the logic used in part 1)
Put the values back in we get your part 2

Only thing now remains rank of (AH) <= min[rank A, rank H]
Which is easy if we consider AH as linear combination of column vectors of A OR equivalently linear combination of row vectors of H.

Hope it helps