Suppose $BA$ is nilpotent, is it always true that $AB$ is nilpotent?Nilpotent matrices with same minimal polynomial and nullityIs the zero matrix the only symmetric, nilpotent matrix with real values?Smoothness of a MatrixSuppose $A^2B+BA^2=2ABA$.Prove that there exists a positive integer $k$ such that $(AB-BA)^k=0$.Product of nilpotent matrices is nilpotentProduct of nilpotent matrices invertibleExample of non-nilpotent product of nilpotent matricesUnitriangular group $UT_n(Bbb Z)$ is nilpotent with class $n$Proving an operator is nilpotentNilpotent matrix sets

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Suppose $BA$ is nilpotent, is it always true that $AB$ is nilpotent?


Nilpotent matrices with same minimal polynomial and nullityIs the zero matrix the only symmetric, nilpotent matrix with real values?Smoothness of a MatrixSuppose $A^2B+BA^2=2ABA$.Prove that there exists a positive integer $k$ such that $(AB-BA)^k=0$.Product of nilpotent matrices is nilpotentProduct of nilpotent matrices invertibleExample of non-nilpotent product of nilpotent matricesUnitriangular group $UT_n(Bbb Z)$ is nilpotent with class $n$Proving an operator is nilpotentNilpotent matrix sets













2












$begingroup$


Given matrices $A$ and $B$ such that $BA$ is nilpotent. Is it true that $AB$ is nilpotent?



My thoughts so far:



$BA$ nilpotent $Rightarrow (BA)^k = 0$ for some positive integer k.



$(BA)^k = (BA)(BA)....(BA) = BABA...BA = B(AB)(AB)....(AB)A = B(AB)^k-1A = 0$



But I have no idea how to proceed from here. If its not true, is there any counterexample?










share|cite|improve this question











$endgroup$
















    2












    $begingroup$


    Given matrices $A$ and $B$ such that $BA$ is nilpotent. Is it true that $AB$ is nilpotent?



    My thoughts so far:



    $BA$ nilpotent $Rightarrow (BA)^k = 0$ for some positive integer k.



    $(BA)^k = (BA)(BA)....(BA) = BABA...BA = B(AB)(AB)....(AB)A = B(AB)^k-1A = 0$



    But I have no idea how to proceed from here. If its not true, is there any counterexample?










    share|cite|improve this question











    $endgroup$














      2












      2








      2





      $begingroup$


      Given matrices $A$ and $B$ such that $BA$ is nilpotent. Is it true that $AB$ is nilpotent?



      My thoughts so far:



      $BA$ nilpotent $Rightarrow (BA)^k = 0$ for some positive integer k.



      $(BA)^k = (BA)(BA)....(BA) = BABA...BA = B(AB)(AB)....(AB)A = B(AB)^k-1A = 0$



      But I have no idea how to proceed from here. If its not true, is there any counterexample?










      share|cite|improve this question











      $endgroup$




      Given matrices $A$ and $B$ such that $BA$ is nilpotent. Is it true that $AB$ is nilpotent?



      My thoughts so far:



      $BA$ nilpotent $Rightarrow (BA)^k = 0$ for some positive integer k.



      $(BA)^k = (BA)(BA)....(BA) = BABA...BA = B(AB)(AB)....(AB)A = B(AB)^k-1A = 0$



      But I have no idea how to proceed from here. If its not true, is there any counterexample?







      linear-algebra






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 2 days ago









      Asaf Karagila

      307k33440773




      307k33440773










      asked 2 days ago









      aaaaaa

      374




      374




















          3 Answers
          3






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          4












          $begingroup$

          $(BA)^k = (BA)(BA)....(BA) = BABA...BA =0$



          Multiply from the left by $A$ and from the right by $B$, you'll get $$ A.BABA...BA.B= (AB)(AB)...(AB)=(AB)^k+1=0$$






          share|cite|improve this answer









          $endgroup$




















            5












            $begingroup$

            You just have to replace $A$ by $B$ in your computation. Then $(AB)^k +1= A(BA)^k B$. I assume you can finish from here.






            share|cite|improve this answer









            $endgroup$




















              2












              $begingroup$

              You can multiply by A on the left side and B on the right side, and you get $(AB)^k+1 = 0$






              share|cite|improve this answer









              $endgroup$













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                3 Answers
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                3 Answers
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                active

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                4












                $begingroup$

                $(BA)^k = (BA)(BA)....(BA) = BABA...BA =0$



                Multiply from the left by $A$ and from the right by $B$, you'll get $$ A.BABA...BA.B= (AB)(AB)...(AB)=(AB)^k+1=0$$






                share|cite|improve this answer









                $endgroup$

















                  4












                  $begingroup$

                  $(BA)^k = (BA)(BA)....(BA) = BABA...BA =0$



                  Multiply from the left by $A$ and from the right by $B$, you'll get $$ A.BABA...BA.B= (AB)(AB)...(AB)=(AB)^k+1=0$$






                  share|cite|improve this answer









                  $endgroup$















                    4












                    4








                    4





                    $begingroup$

                    $(BA)^k = (BA)(BA)....(BA) = BABA...BA =0$



                    Multiply from the left by $A$ and from the right by $B$, you'll get $$ A.BABA...BA.B= (AB)(AB)...(AB)=(AB)^k+1=0$$






                    share|cite|improve this answer









                    $endgroup$



                    $(BA)^k = (BA)(BA)....(BA) = BABA...BA =0$



                    Multiply from the left by $A$ and from the right by $B$, you'll get $$ A.BABA...BA.B= (AB)(AB)...(AB)=(AB)^k+1=0$$







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered 2 days ago









                    Fareed AFFareed AF

                    692112




                    692112





















                        5












                        $begingroup$

                        You just have to replace $A$ by $B$ in your computation. Then $(AB)^k +1= A(BA)^k B$. I assume you can finish from here.






                        share|cite|improve this answer









                        $endgroup$

















                          5












                          $begingroup$

                          You just have to replace $A$ by $B$ in your computation. Then $(AB)^k +1= A(BA)^k B$. I assume you can finish from here.






                          share|cite|improve this answer









                          $endgroup$















                            5












                            5








                            5





                            $begingroup$

                            You just have to replace $A$ by $B$ in your computation. Then $(AB)^k +1= A(BA)^k B$. I assume you can finish from here.






                            share|cite|improve this answer









                            $endgroup$



                            You just have to replace $A$ by $B$ in your computation. Then $(AB)^k +1= A(BA)^k B$. I assume you can finish from here.







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered 2 days ago









                            Severin SchravenSeverin Schraven

                            6,7802936




                            6,7802936





















                                2












                                $begingroup$

                                You can multiply by A on the left side and B on the right side, and you get $(AB)^k+1 = 0$






                                share|cite|improve this answer









                                $endgroup$

















                                  2












                                  $begingroup$

                                  You can multiply by A on the left side and B on the right side, and you get $(AB)^k+1 = 0$






                                  share|cite|improve this answer









                                  $endgroup$















                                    2












                                    2








                                    2





                                    $begingroup$

                                    You can multiply by A on the left side and B on the right side, and you get $(AB)^k+1 = 0$






                                    share|cite|improve this answer









                                    $endgroup$



                                    You can multiply by A on the left side and B on the right side, and you get $(AB)^k+1 = 0$







                                    share|cite|improve this answer












                                    share|cite|improve this answer



                                    share|cite|improve this answer










                                    answered 2 days ago









                                    GatgatGatgat

                                    1775




                                    1775



























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