Results 1 to 4 of 4

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

  1. #1
    Newbie
    Joined
    Sep 2009
    Posts
    7

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

    I need to prove that

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

    When H is nonsingular nxn, and A is mxn.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Apr 2009
    Posts
    677
    Quote Originally Posted by be more View Post
    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.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2009
    Posts
    7
    Quote Originally Posted by aman_cc View Post
    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.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Apr 2009
    Posts
    677
    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
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Row Rank = Column Rank?
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 9th 2011, 12:10 AM
  2. Proof: rank(AB)+n >= rank(A)+rank(B)
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: September 9th 2010, 05:28 PM
  3. Replies: 3
    Last Post: August 20th 2010, 05:32 AM
  4. Row rank and column rank
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: April 13th 2010, 07:40 AM
  5. Short proof that rows-rank=column-rank?
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: June 26th 2009, 10:02 AM

Search Tags


/mathhelpforum @mathhelpforum