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Another question on ranks (linear algebra).

  1. Feb 5, 2007 #1
    i need to prove that for A square matrix: rank AA^t=rank A.
    well rank AA^t<=rank A, but how do i show that rankAA^t>=rankA, i mean i need to show if x is a solution of AA^tx=0 then x is a solution of Ax=0 or of A^tx=0, but how?
     
  2. jcsd
  3. Feb 5, 2007 #2
    You are trying to show that the transpose of an nxn multiplied by itself has the same or less rank? I would use the definition of matrix multiplication, and then transpose, and do Gauss elimination (probably on a 2x2). There may be a more subtle way, but the forceful approach should work.
     
  4. Feb 5, 2007 #3

    radou

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    Let U = {x : AA^t x = 0}, and V = {x : A^t x = 0}. Assume you have already shown that dim V <= dim U (which implies r(A) >= r(AA^t) ), and now assume dim U > dim V (which would imply r(A) > r(AA^t) ) and find a counterexample (it's simple). When you have shown that dim U > dim V can't hold, then dim U = dim V must hold, and hence r(A^t) = r(A) = r(AA^t). Hope this works.
     
  5. Feb 5, 2007 #4
    i dont think this would work, cause in ad absurdum proofs you need to get a logical contradiction, not a counter example, perhaps im wrong here and you are right, but i dont think so.
     
  6. Feb 5, 2007 #5

    radou

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    Actually, that's what's bothering me, too. If we assume it holds for any matrix, could a 'proof by counterexample' work? Guess some of the PF mathematicians should take over on this one.
     
  7. Feb 5, 2007 #6

    AlephZero

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    You can show this directly from the singular value decomposition of A.

    Let the SVD of A be U.D.V^T where D is the diagonal matrix of singular values. Write the SVD of A.A^T in terms of U D and V, and show that A and A.A^T have the same number of zero singular values, hence the same rank.

    There is a similar proof involving eigenvalues and eigenvectors.

    There is probably a similar but more "elementary" proof involving choosing a set of basis vectors where a subset of the basis vectors spans the null space of A - but I haven't thought idea that right through.
     
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