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Math Help - Find all matrices {(a,b);(c,d)} that will commute with every matrix in set S

  1. #1
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    Find all matrices {(a,b);(c,d)} that will commute with every matrix in set S

    where set S is the set of all 2x2 matrices

    so i set an arbitrary matrix from the set S {(w,x);(y,z)}

    so (sorry i dont know the matrix format :< )

    [a b][w x] = [w x][a b]
    [c d][y z] [y z] [c d]

    and got

    [(ax+bz) (ay+bw)] = [(xa+yc) (xb+yd)]
    [(cx+dz) (cy+dw)] = [(za+wc) (zb+wd)]

    and then equating them but im stuck here, lol xD
    seems like i forgot algebra! XD any help please?
    Last edited by experiment00; August 15th 2010 at 12:26 PM.
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  2. #2
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    so your matrix must commute with e_{11}=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} and e_{12}=\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}. thus ...
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  3. #3
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    hmm, why should they commute with that 2 specific matrices sir? sorry
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  4. #4
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    Well, first you want it to commute with all matrices so you might as well chose some. The more important reason is that it is easy to work with these easy matrices. If you use all four of these matrices with one entry 1, you can build all other matrices as a linear combination of those four. If your matrix commutes with each of the four, then it should commute with all linear combinations of these four.
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