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Homework Help: Matrix rank proof

  1. Oct 4, 2008 #1
    I'm trying to show that any matrix X with rank n can be written as the sum of matrices Z and Y with rank n-1 and 1, respectively.

    Since X,Y, Z have the same dimensions, is this a simple matter of saying pick one of the columns in X with a pivot. Let Z= X with this column replaced by zeroes but all other entries the same as X. Let Y consist of the removed column and all other entries 0. Thus, X=Y+Z and Y has rank n-1 and Z has rank 1.

    Does this look correct?
     
  2. jcsd
  3. Oct 5, 2008 #2

    morphism

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    Yes, that's fine.
     
  4. Oct 5, 2008 #3
    it doesnt seem like this is the most formal logic in the world...is it trivial that the matrices constructed have ranks of n-1 and 1 or does this need to be shown as well
     
  5. Oct 5, 2008 #4

    morphism

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    Since the rank of X is n, its columns are linearly independent. Hence any subset of those columns is linearly independent as well.
     
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