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Proof of symmetric and anti symmetric matrices

  1. Aug 31, 2011 #1
    1. The problem statement, all variables and given/known data

    aij is a symmetric matrix
    bij is a an anti symmetric matrix

    prove that aij * bij = 0


    2. Relevant equations

    aij * bij

    3. The attempt at a solution


    any one got any ideas ?
     
  2. jcsd
  3. Aug 31, 2011 #2

    Hootenanny

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    HINT: What happens when you interchange the indices?
     
  4. Aug 31, 2011 #3

    Fredrik

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    [itex]a_{ij}[/itex] doesn't denote a matrix. It denotes the component on row i, column j, of a matrix.

    Since [tex]\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}=\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}\neq 0,[/tex] it's not true that the product of a symmetric and an antisymmetric matrix is =0. On the other hand, it is true that [itex]a_{ij}b_{ij}=0[/itex] (assuming that repeated indices are summed over). You should take some time to think about what the expression [itex]a_{ij}b_{ij}[/itex] really means, and what matrix operation(s) it involves.

    Do you know the definition of matrix multiplication? If [itex]a_{ij}[/itex] denotes a component of a matrix A, and [itex]b_{ij}[/itex] denotes a component of a matrix B. Then what will you find on row i, column j of AB?
     
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