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Summation notation

  1. Feb 8, 2012 #1
    I am trying to understand summation notation and there are a few inconsistencies in my head that I would like to clear up.

    Suppose C is an m*n matrix and [itex] \vec{x} [/itex] is a 1*m row vector. Then,

    [tex] \vec{x}C = \sum_{i} x_{i}C_{ij} = \sum_{i} C_{ij}x_{i} = \sum_{i} {C_{ji}}^Tx_{i} = C^T \vec{x} [/tex]

    This is clearly wrong but I'm not sure which operation is wrong. In terms of dimensions it doesn't make sense since [itex] C^T [/itex] is n*m and [itex] \vec{x} [/itex] is 1*m.
     
  2. jcsd
  3. Feb 9, 2012 #2
    The discrepancy arises because this way of defining a matrix and vector does not differentiate between (e.g.) a row vector and a column vector. Have a look at this for a general introduction to Einstein's notation and this for it's implementation in your case. No issues there!
     
  4. Feb 9, 2012 #3

    HallsofIvy

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    Specifically, it is saying that
    [tex]\begin{bmatrix}x & y\end{bmatrix}\begin{bmatrix}a & b \\ c & d\end{bmatrix}= \begin{bmatrix}a & c \\ b & d\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}[/tex]
     
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