1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    [iTEX]

    A*A=\sum_{k=1}^{n}a_{ik}a_{kj}

    [/iTEX]



    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

    [TEX]

    tr(A^{2})=\sum_{i=1}^{n}A_{ii}^{2}=\sum_{i=1}^{n}(\sum_{i=1}^{n}a_{ik}a_{ki})

    [/TEX]

    and because the matrix is simmetric then the trace is zero

    why?


    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

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    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

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Ah, you should have said they were symmetric! :smile:

    If

    [tex]A=\left(\begin{array}{ccccc}
    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}\\
    \end{array}\right)[/tex]

    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
    etc..
    so i get this expression
    [tex]tr(A^{2})=\sum_{k=1}^{n}\sum_{i=1}^{n}a_{ki}a_{ik}[/tex]
    so because its simetric and equlas zero
    [tex]tr(A^{2})=\sum_{k=1}^{n}\sum_{i=1}^{n}(a_{ki})^2=0[/tex]
    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

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Indeed, so the k'th member in the diagonal is

    [tex]\|(a_{1k},...,a_{nk})\|^2[/tex]

    Do you see that?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Trace prove question
  1. Trace question (Replies: 2)

  2. Prove question (Replies: 9)

Loading...