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Sumation of symmetric and skew symmetri metrices

  1. Jul 2, 2011 #1
    Express \left(\begin{array}{cccc}
    6 & 1 & 5\\
    -2 & -5 & 4\\
    -3 & 3 & -1\
    end{array}
    \right) as the sum of the symmetric and skew symmetric matrices.

    I did this following way

    Consider symmetric metric as "A"
    then;
    A = \left(\begin{array}{cccc}
    6 & 1 & 5\\
    1 & -5 & 4\\
    5 & 4 & -1\
    \end{array}
    \right)

    Consider skew symmetric metric as "B"
    then;
    B = \left(\begin{array}{cccc}
    0 & 1 & 5\\
    -1 & 0 & 4\\
    -5 & -4 & 0\
    \end{array}
    \right)

    Then sum of matrices A and B is;
    A+B= \left(\begin{array}{cccc}
    6 & 2 & 10\\
    0 & -5 & 8\\
    0 & 0 & -1\
    \end{array}
    \right)

    is this correct??:smile:
     
  2. jcsd
  3. Jul 2, 2011 #2

    vela

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    Fixing your LaTeX...
    No. The problem is asking you to find A and B such that A+B is equal to the original matrix. This is obviously not the case for your A and B.
     
  4. Jul 2, 2011 #3
    Thank you for fixing Latex vela :smile:, oh.. I think I've understood the question wrongly, so can you give me a hint, on how to do that in correct way :smile:
     
  5. Jul 2, 2011 #4

    I like Serena

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    Let's take a generic symmetric and a skewed symmetric matrix.

    Say:
    [tex]A=\begin{pmatrix}a & b \\ b & d \end{pmatrix}\qquad A=\begin{pmatrix}0 & q \\ -q & 0\end{pmatrix}[/tex]

    Adding them up will yield:
    [tex]A+B=\begin{pmatrix}a & b+q \\ b-q & d \end{pmatrix}[/tex]

    You should note that the average of (b-q) and (b+q) is b.

    Now can you think up how to construct a symmetric and a skewed symmetric matrix from a given matrix?
     
  6. Jul 2, 2011 #5
    Thank you I like Serena, I've found a formula to express a square matrix by using symmetric and skew symmetric matrices here it is;

    Let A be the given square matrix
    A can be uniquely expressed as sum of a symmetric matrix and a skew symmetric matrix, which is

    A =(A+A')/2 + (A-A')/2 consider A' is Transpose of matrix A;
    by using this I was able to got the symmetric matrix and a skew symmetric matrix for the given matrix.:smile:. is this correct?
     
  7. Jul 2, 2011 #6

    I like Serena

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    Yes, this this correct.
     
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