nested commutators

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I have defined the commutator between two matrices as



comm[A_,B_]:=A.B-B.A


I have defined the nested commutator as 



nestcomm[A_,B_,n_]:= ToExpression[ 
 StringRepeat["comm[a,", n] <> "b" <> StringRepeat["]", n] 
 ]


where $n$ indicates how many time the commutator must be nested.



Does exists a more simple way to define an $n$-time nested operation between two elements $A$ and $B$? I tried to use the Nest function but without success.






nest







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    up vote
    1
    down vote

    favorite












    I have defined the commutator between two matrices as



    comm[A_,B_]:=A.B-B.A


    I have defined the nested commutator as 



    nestcomm[A_,B_,n_]:= ToExpression[ 
     StringRepeat["comm[a,", n] <> "b" <> StringRepeat["]", n] 
     ]


    where $n$ indicates how many time the commutator must be nested.



    Does exists a more simple way to define an $n$-time nested operation between two elements $A$ and $B$? I tried to use the Nest function but without success.






    nest







    share|improve this question























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      I have defined the commutator between two matrices as



      comm[A_,B_]:=A.B-B.A


      I have defined the nested commutator as 



      nestcomm[A_,B_,n_]:= ToExpression[ 
       StringRepeat["comm[a,", n] <> "b" <> StringRepeat["]", n] 
       ]


      where $n$ indicates how many time the commutator must be nested.



      Does exists a more simple way to define an $n$-time nested operation between two elements $A$ and $B$? I tried to use the Nest function but without success.






      nest







      share|improve this question













      I have defined the commutator between two matrices as



      comm[A_,B_]:=A.B-B.A


      I have defined the nested commutator as 



      nestcomm[A_,B_,n_]:= ToExpression[ 
       StringRepeat["comm[a,", n] <> "b" <> StringRepeat["]", n] 
       ]


      where $n$ indicates how many time the commutator must be nested.



      Does exists a more simple way to define an $n$-time nested operation between two elements $A$ and $B$? I tried to use the Nest function but without success.






      nest




      nest






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      share|improve this question










      asked 5 hours ago









      Galuoises

      1887




      1887




















          2 Answers
          2






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          up vote
          4
          down vote



          accepted










          Redefine your comm function to take in and put out a pair of matrices, then Nest can be called easily:



          comm[A_, B_] := A, A.B - B.A


          Then just invoke Nest with the number of terms in the final position. For n=2:



          Rest[Nest[comm, a, b, 2]]
          a.(a.b - b.a) - (a.b - b.a).a





          share|improve this answer




















          • Thank you for your help!
            – Galuoises
            6 mins ago

















          up vote
          2
          down vote













          Would something like the following work?



          comm[A_,B_,1] := A.B-B.A
          comm[A_, B_, n_Integer?Positive] := comm[comm[A, B, n-1], B, 1]


          For example:



          comm[A, B, 2] //TensorExpand
          comm[A, B, 3] //TensorExpand



          A.B.B - 2 B.A.B + B.B.A



          A.B.B.B - 3 B.A.B.B + 3 B.B.A.B - B.B.B.A




          Use:



          comm[A_, B_, n_Integer?Positive] := comm[A, comm[A, B, n-1], 1]


          to obtain the same definition as in @bill's answer.






          share|improve this answer




















          • Thank you! I appreciate your answer
            – Galuoises
            7 mins ago










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          2 Answers
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          active

          oldest

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          2 Answers
          2






          active

          oldest

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          active

          oldest

          votes








          up vote
          4
          down vote



          accepted










          Redefine your comm function to take in and put out a pair of matrices, then Nest can be called easily:



          comm[A_, B_] := A, A.B - B.A


          Then just invoke Nest with the number of terms in the final position. For n=2:



          Rest[Nest[comm, a, b, 2]]
          a.(a.b - b.a) - (a.b - b.a).a





          share|improve this answer




















          • Thank you for your help!
            – Galuoises
            6 mins ago














          up vote
          4
          down vote



          accepted










          Redefine your comm function to take in and put out a pair of matrices, then Nest can be called easily:



          comm[A_, B_] := A, A.B - B.A


          Then just invoke Nest with the number of terms in the final position. For n=2:



          Rest[Nest[comm, a, b, 2]]
          a.(a.b - b.a) - (a.b - b.a).a





          share|improve this answer




















          • Thank you for your help!
            – Galuoises
            6 mins ago












          up vote
          4
          down vote



          accepted







          up vote
          4
          down vote



          accepted






          Redefine your comm function to take in and put out a pair of matrices, then Nest can be called easily:



          comm[A_, B_] := A, A.B - B.A


          Then just invoke Nest with the number of terms in the final position. For n=2:



          Rest[Nest[comm, a, b, 2]]
          a.(a.b - b.a) - (a.b - b.a).a





          share|improve this answer












          Redefine your comm function to take in and put out a pair of matrices, then Nest can be called easily:



          comm[A_, B_] := A, A.B - B.A


          Then just invoke Nest with the number of terms in the final position. For n=2:



          Rest[Nest[comm, a, b, 2]]
          a.(a.b - b.a) - (a.b - b.a).a






          share|improve this answer












          share|improve this answer



          share|improve this answer










          answered 5 hours ago









          bill s

          51.8k375146




          51.8k375146











          • Thank you for your help!
            – Galuoises
            6 mins ago
















          • Thank you for your help!
            – Galuoises
            6 mins ago















          Thank you for your help!
          – Galuoises
          6 mins ago




          Thank you for your help!
          – Galuoises
          6 mins ago










          up vote
          2
          down vote













          Would something like the following work?



          comm[A_,B_,1] := A.B-B.A
          comm[A_, B_, n_Integer?Positive] := comm[comm[A, B, n-1], B, 1]


          For example:



          comm[A, B, 2] //TensorExpand
          comm[A, B, 3] //TensorExpand



          A.B.B - 2 B.A.B + B.B.A



          A.B.B.B - 3 B.A.B.B + 3 B.B.A.B - B.B.B.A




          Use:



          comm[A_, B_, n_Integer?Positive] := comm[A, comm[A, B, n-1], 1]


          to obtain the same definition as in @bill's answer.






          share|improve this answer




















          • Thank you! I appreciate your answer
            – Galuoises
            7 mins ago














          up vote
          2
          down vote













          Would something like the following work?



          comm[A_,B_,1] := A.B-B.A
          comm[A_, B_, n_Integer?Positive] := comm[comm[A, B, n-1], B, 1]


          For example:



          comm[A, B, 2] //TensorExpand
          comm[A, B, 3] //TensorExpand



          A.B.B - 2 B.A.B + B.B.A



          A.B.B.B - 3 B.A.B.B + 3 B.B.A.B - B.B.B.A




          Use:



          comm[A_, B_, n_Integer?Positive] := comm[A, comm[A, B, n-1], 1]


          to obtain the same definition as in @bill's answer.






          share|improve this answer




















          • Thank you! I appreciate your answer
            – Galuoises
            7 mins ago












          up vote
          2
          down vote










          up vote
          2
          down vote









          Would something like the following work?



          comm[A_,B_,1] := A.B-B.A
          comm[A_, B_, n_Integer?Positive] := comm[comm[A, B, n-1], B, 1]


          For example:



          comm[A, B, 2] //TensorExpand
          comm[A, B, 3] //TensorExpand



          A.B.B - 2 B.A.B + B.B.A



          A.B.B.B - 3 B.A.B.B + 3 B.B.A.B - B.B.B.A




          Use:



          comm[A_, B_, n_Integer?Positive] := comm[A, comm[A, B, n-1], 1]


          to obtain the same definition as in @bill's answer.






          share|improve this answer












          Would something like the following work?



          comm[A_,B_,1] := A.B-B.A
          comm[A_, B_, n_Integer?Positive] := comm[comm[A, B, n-1], B, 1]


          For example:



          comm[A, B, 2] //TensorExpand
          comm[A, B, 3] //TensorExpand



          A.B.B - 2 B.A.B + B.B.A



          A.B.B.B - 3 B.A.B.B + 3 B.B.A.B - B.B.B.A




          Use:



          comm[A_, B_, n_Integer?Positive] := comm[A, comm[A, B, n-1], 1]


          to obtain the same definition as in @bill's answer.







          share|improve this answer












          share|improve this answer



          share|improve this answer










          answered 5 hours ago









          Carl Woll

          62.4k281158




          62.4k281158











          • Thank you! I appreciate your answer
            – Galuoises
            7 mins ago
















          • Thank you! I appreciate your answer
            – Galuoises
            7 mins ago















          Thank you! I appreciate your answer
          – Galuoises
          7 mins ago




          Thank you! I appreciate your answer
          – Galuoises
          7 mins ago

















           

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