Commutation property

  • Thread starter vdgreat
  • Start date
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..
 

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.
 

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.
 
but how can i prove it or generalize it??
 
help with this problem

anyone?
 

morphism

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Did you think about what has been posted already?
 

HallsofIvy

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Which of the basis matrices I gave commute with all other matrices?
 
try to look over schur lemma... it is a generalztion of what you asked....

ciao
marco
 
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.
 
i got it thanks
 

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