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Math Help - Matrix eqn

  1. #1
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    Matrix eqn

    Matrix A = (3 -5 1 -1
    4 -1 2 0
    0 2 -1 -1
    3 0 -3 1)


    Express A as the sum of a symmetric matrix U and a skew-symmetric matrix V such that A = U+V

    Two unkowns in this Q? How can i solve this?
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  2. #2
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    Quote Originally Posted by MilK View Post
    Matrix A = (3 -5 1 -1
    4 -1 2 0
    0 2 -1 -1
    3 0 -3 1)


    Express A as the sum of a symmetric matrix U and a skew-symmetric matrix V such that A = U+V

    Two unkowns in this Q? How can i solve this?
    You are looking for the solution of
    \left [ \begin{matrix} 3 & -5 & 1 & -1 \\ 4 & -1 & 2 & 0 \\0 & 2 & -1 & -1 \\ 3 & 0 & -3 & 1 \end{matrix} \right ] = \left [ \begin{matrix} a & b & c & d \\ b & e & f & g \\c & f & h & i \\ d & g & i & j \end{matrix} \right ] + \left [ \begin{matrix} 0 & k & m & n \\ -k & 0 & p & q \\-m & -p & 0 & r \\ -n & -q & -r & 0 \end{matrix} \right ]

    a, e, h, and j are easy. For the others you have something like:
    -5 = b + k
    and
    4 = b - k
    Solve for b and k.

    It'll take a while, but it can be done.

    Or perhaps you will simply want to calculate
    U = \frac{1}{2} \cdot \left ( A + A^T \right )
    and
    V = \frac{1}{2} \cdot \left ( A - A^T \right )

    -Dan
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