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    How to show that: A A^T = \displaystyle \sum_i a_i a_i^T, where A is symmetric matrix?

    Example:

    When A=\left[\begin{array}{ccc}1&2&3\\2&3&1\\  3&1&2\end{array}\right] then

    A A^T = \left[ \begin{array}{ccc} 1\\2\\3  \end{array} \right] \left[ \begin{array}{ccc} 1&2&3 \end{array}  \right] + \left[ \begin{array}{ccc} 2\\3\\1 \end{array} \right] \left[  \begin{array}{ccc} 2&3&1 \end{array} \right] + \left[  \begin{array}{ccc} 3\\1\\2 \end{array} \right] \left[ \begin{array}{ccc}  3&1&2 \end{array} \right]

    So a_i is the column vector (i-th column).
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    If A is symmetric then A = A^T and A is an n \times n matrix.

    AA^T = (A_{ij})^2 = \displaystyle\sum_{k=1}^n a_{ik}a_{kj}

    From the definition of matrix multiplication. Do you see how to finish it?
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    Thank you for reply.

    \displaystyle\sum_{k=1}^n a_{ik}a_{kj} = \displaystyle\sum_{k=1}^n a_{ik}a_{jk}^T

    So now can I rewrite as: \displaystyle\sum_{k=1}^n a_k a_k^T or would it be incorrect?
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    (Original post by Noble.)
    If A is symmetric then A = A^T and A is an n \times n matrix.

    AA^T = (A_{ij})^2 = \displaystyle\sum_{k=1}^n a_{ik}a_{kj}

    From the definition of matrix multiplication. Do you see how to finish it?
    I am being a pedant here but you should really write something like

    \left(\displaystyle\sum_{k=1}^n a_{ik}a_{kj}\right)

    since the inside of the bracket is just a component; not a matrix.
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    (Original post by Mark85)
    I am being a pedant here but you should really write something like

    \left(\displaystyle\sum_{k=1}^n a_{ik}a_{kj}\right)

    since the inside of the bracket is just a component; not a matrix.
    Interesting. They don't do this in the lecture notes.
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    (Original post by Noble.)
    Interesting. They don't do this in the lecture notes.
    Well, you wrote brackets round the A_{ij} which is just the i,jth component of A. It doesn't matter whether you choose to use brackets around a typical component to signify you mean the whole matrix or not but you should at least be consistent.

    I mean, sure it is very clear what is meant but such things might cause a problem for people learning e.g. the OP.
 
 
 
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