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Commutation property

  1. Mar 20, 2008 #1
    Let D =

    [d11 d12]
    [d21 d22]

    be a 2x2 matrix. Prove that D commutes with all other 2x2
    matrices if and only if d12 = d21 = 0 and d11 = d22.

    I know if we can prove for every A, AD=DA should be true, but I really dont know how to proceed from there. I tried equating elements of AD with DA but that really didnt help.

    Can anyone help me with this problem. Thanks..
     
  2. jcsd
  3. Mar 20, 2008 #2

    morphism

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    Try particular As, like

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

    and see where that leads you.
     
  4. Mar 20, 2008 #3

    HallsofIvy

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    morphism's idea is excellent. Do you see where he got it?
    The matrices
    [tex]\left(\begin{array}{cc}1 & 0 \\ 0 & 0 \end{array}\right)[/tex]
    [tex]\left(\begin{array}{cc}0 & 1 \\ 0 & 0 \end{array}\right)[/tex]
    [tex]\left(\begin{array}{cc}0 & 0 \\ 1 & 0 \end{array}\right)[/tex]
    [tex]\left(\begin{array}{cc}0 & 0 \\ 0 & 1 \end{array}\right)[/tex]
    form a basis for the vector space of all 2 by 2 matrices. What is true for the basis is true for all 2 by 2 matrices.
     
  5. Mar 21, 2008 #4
    but how can i prove it or generalize it??
     
  6. Mar 24, 2008 #5
    help with this problem

    anyone?
     
  7. Mar 24, 2008 #6

    morphism

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    Did you think about what has been posted already?
     
  8. Mar 24, 2008 #7

    HallsofIvy

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    Which of the basis matrices I gave commute with all other matrices?
     
  9. Mar 24, 2008 #8
    try to look over schur lemma... it is a generalztion of what you asked....

    ciao
    marco
     
  10. Mar 24, 2008 #9
    for each of the above matrices, i found out that it is true. but how can i prove this without having knowledge of basis. i haven't dont it yet.
     
  11. Mar 27, 2008 #10
    i got it thanks
     
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