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Homework Help: Trace prove question

  1. Jun 21, 2011 #1
    i need to prove that if tr(A^2)=0

    then A=0

    we have a multiplication of 2 the same simmetrical matrices

    why there multiplication is this sum formula




    i know that wjen we multiply two matrices then in our result matrix

    each aij member is dot product of i row and j column.

    dont understand the above formula.

    and i dont understand how they got the following formula:

    then when we calculate the trace (the sum of the diagonal members)

    we get




    and because the matrix is simmetric then the trace is zero


    i need to prove that if tr(A^2)=0

    then A=0

    can you explain the sigma work in order to prove it?
  2. jcsd
  3. Jun 21, 2011 #2
    Hi nhrock3! :smile:

    This is false. Consider

    [tex]A=\left(\begin{array}{cc} 0 & 1\\ 0 & 0\\ \end{array}\right)[/tex]

    then tr(A2)=0, but A is not zero.

    What does the exercise say precisely?
  4. Jun 21, 2011 #3
    if A is a simetric matrix and if tr(A^2)=0 then A=0
  5. Jun 21, 2011 #4
    Ah, you should have said they were symmetric! :smile:


    a_{11} & a_{12} & a_{13} & ... & a_{1n}\\
    a_{12} & a {22} & a_{23} & ... & a_{2n}\\
    a_{13} & a_{23} & a_{33} & ... & a_{3n}\\
    \vdots & \vdots & \vdots & \ddots & \vdots\\
    a_{1n} & a_{2n} & a_{3n} & ... & a_{nn}\\

    then what will be the diagonal of A2?? In particular, can you show that the diagonal contains only positive values?
  6. Jun 21, 2011 #5
    each member of the diagonal on the A^2 matrix its number of row equals the column number

    so the second member in the diagonal is the dot product of the second row with the second column
    so i get this expression
    so because its simetric and equlas zero
    so we get a sum of squeres and in order for them to be zero
    then each one of them has to be zero
    thannkkks :)
    Last edited: Jun 21, 2011
  7. Jun 21, 2011 #6
    Indeed, so the k'th member in the diagonal is


    Do you see that?
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