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The Transpose of a Matrix

  1. Sep 26, 2013 #1
    Let [itex]B=A^{T}A[/itex]. Show that [itex]b_{ij}[/itex]=[itex]a^{T}_{i}[/itex][itex]a_{j}[/itex].

    I have no idea how to approach this problem.
     
  2. jcsd
  3. Sep 26, 2013 #2
    I don't understand your notation. Could you please clarify what you've written?
     
  4. Sep 26, 2013 #3
    Let [itex]B=A^{T}A[/itex]. Show that [itex]b_{ij}[/itex]=[itex]a^{T}_{i}[/itex][itex]a_{j}[/itex].

    When you transpose [itex]A[/itex]You are flipping the rows and columns.
    When you multiply [itex]A^{T}A[/itex] you would generate each element [itex]b_{ij}[/itex] will be the dot product of row i of the first matrix and column j of the second matrix. Since the first matrix is the transposition of the second, row i of that transposition will be column i of the original. So each element will be the dot product of two column vectors:

    [itex]b_{ij}[/itex]=[itex]a_{i}[/itex]·[itex]a_{j}[/itex]

    Using matrix notation, this is:

    [itex]b_{ij}[/itex]=[itex]a^{T}_{i}[/itex][itex]a_{j}[/itex]
     
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