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Homework Help: Linear Algebra - Matrix Symitry

  1. Jul 22, 2012 #1
    1. The problem statement, all variables and given/known data

    Code (Text):
    A matrix A is skew-symmetric if A[SUP]T[/SUP] = -A. Write the matrix
    B below as the sum of a symmetric matrix and a skew-symmetric matrix.
    B =  a b c
         d e f
         g h i
    3. The attempt at a solution

    So I'm Pretty sure that the
    Symetric Matrix = B + BT
    Skew Symetric Martix = B - BT

    So B+BT+ B - BT should equal B but I get 2 B.

    So B = 1/2((B+BT) + (B - BT))

    Code (Text):

    B[SUP]T[/SUP] = a d g
         b e h
         c f i

    Sym = B + B[SUP]T[/SUP]
        =  2a b+d c+g
          d+b  2e f+h
          g+c h+f  2i

    Skew Sym = B - B[SUP]T[/SUP]
        =  0  b-d c-g
          d-b  0  f-h
          g-c h-f  0

    So If you add Those... You get

    2a 2b 2c = 2B
    2d 2e 2f
    2g 2h 2i
    Then divide by 2 and I'm done right?
  2. jcsd
  3. Jul 22, 2012 #2


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    Staff Emeritus
    Science Advisor
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    Gold Member

    Actually, you're pretty close to getting a correct result.

    The symmetric matrix you need is (1/2)(B + BT), the skew-symmetric matrix is (1/2)(B - BT) .
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