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Showing a Matrix is Symmetric

  1. Oct 25, 2008 #1
    1. The problem statement, all variables and given/known data

    A matrix S is symmetric if S = ST. Show that AAT is symmetric for any matrix A.

    2. Relevant equations

    AAT = (AT)TAT

    3. The attempt at a solution

    I just said:

    A = (AT)T and AT = AT

    Therefore AAT is symmetric.

    I am unsure if that proves it or if I just went in a circle proving nothing.
     
  2. jcsd
  3. Oct 25, 2008 #2

    statdad

    User Avatar
    Homework Helper

    If [tex] A, B [/tex] are any matrices, how do you simplify the following expression?

    [tex]
    \left(AB\right)^T
    [/tex]

    Apply that answer to simplify

    [tex]
    A A^T
    [/tex]

    and use the hints you provided
     
  4. Oct 25, 2008 #3
    Ok I think this is right.

    (AB)T = BTAT

    Then (AAT)T = AAT

    Therefore AAT is symmetric because (AT)T = A and AT = AT.
     
  5. Oct 25, 2008 #4

    statdad

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    Homework Helper

    I think you have the correct idea. the 'conventional' way of finishing the proof would be to write everything in one string rather than stopping midstream - I have no idea how
    picky your professor would be in that regard.
     
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