Noncommuting complex matrices: Existence of a simultaneous eigenvector












3














Let $A$ and $B$ be $ntimes n$ matrices with complex entries such that
$AB - BA$ is a linear combination of $A$ and $B$.



I'd like to prove that there exists a non-zero vector $v$ that is an eigenvector of both $A$ and $B$.










share|cite|improve this question
























  • Anything you tried to solve this problem?
    – Hetebrij
    Nov 29 '15 at 22:17










  • So, I know that eigenvectors are the solutions such that the determinant is zero when you know the eigenvalues to each. I assume that, since the matrices are distinct, their eigenvalues will have to be different for that matching eigenvector. Otherwise, I am very lost about why the linear combination of A and B is important.
    – Mark Frazier
    Nov 29 '15 at 22:27












  • What is given means $AB-BA=lambda A+mu B$. Use that
    – Denis Düsseldorf
    Nov 29 '15 at 22:33










  • The linear combination of $A$ and $B$ is important because in general we do not know if $A$ and $B$ are diagonalizable w.r.t. the same basis. However, in the more general case that $AB-BA=0$, we do know that $A$ and $B$ are diagonalizable w.r.t. the same basis.
    – Hetebrij
    Nov 29 '15 at 23:41












  • In fact $A,B$ are simultaneously triangularizable. If you know Lie theory, then it is very easy. Otherwise do as follows. 1. Reduce the problem to the case $AB-BA=A$. 2. Calculate, by a recurrence reasoning, $A^kB-BA^k$. 3. Show that $A$ is nilpotent. 4. Consider $ker(A)$.
    – loup blanc
    Dec 3 '15 at 15:13
















3














Let $A$ and $B$ be $ntimes n$ matrices with complex entries such that
$AB - BA$ is a linear combination of $A$ and $B$.



I'd like to prove that there exists a non-zero vector $v$ that is an eigenvector of both $A$ and $B$.










share|cite|improve this question
























  • Anything you tried to solve this problem?
    – Hetebrij
    Nov 29 '15 at 22:17










  • So, I know that eigenvectors are the solutions such that the determinant is zero when you know the eigenvalues to each. I assume that, since the matrices are distinct, their eigenvalues will have to be different for that matching eigenvector. Otherwise, I am very lost about why the linear combination of A and B is important.
    – Mark Frazier
    Nov 29 '15 at 22:27












  • What is given means $AB-BA=lambda A+mu B$. Use that
    – Denis Düsseldorf
    Nov 29 '15 at 22:33










  • The linear combination of $A$ and $B$ is important because in general we do not know if $A$ and $B$ are diagonalizable w.r.t. the same basis. However, in the more general case that $AB-BA=0$, we do know that $A$ and $B$ are diagonalizable w.r.t. the same basis.
    – Hetebrij
    Nov 29 '15 at 23:41












  • In fact $A,B$ are simultaneously triangularizable. If you know Lie theory, then it is very easy. Otherwise do as follows. 1. Reduce the problem to the case $AB-BA=A$. 2. Calculate, by a recurrence reasoning, $A^kB-BA^k$. 3. Show that $A$ is nilpotent. 4. Consider $ker(A)$.
    – loup blanc
    Dec 3 '15 at 15:13














3












3








3







Let $A$ and $B$ be $ntimes n$ matrices with complex entries such that
$AB - BA$ is a linear combination of $A$ and $B$.



I'd like to prove that there exists a non-zero vector $v$ that is an eigenvector of both $A$ and $B$.










share|cite|improve this question















Let $A$ and $B$ be $ntimes n$ matrices with complex entries such that
$AB - BA$ is a linear combination of $A$ and $B$.



I'd like to prove that there exists a non-zero vector $v$ that is an eigenvector of both $A$ and $B$.







linear-algebra matrices eigenvalues-eigenvectors lie-algebras






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 5 at 7:28









Hanno

2,061425




2,061425










asked Nov 29 '15 at 22:13









Mark FrazierMark Frazier

325




325












  • Anything you tried to solve this problem?
    – Hetebrij
    Nov 29 '15 at 22:17










  • So, I know that eigenvectors are the solutions such that the determinant is zero when you know the eigenvalues to each. I assume that, since the matrices are distinct, their eigenvalues will have to be different for that matching eigenvector. Otherwise, I am very lost about why the linear combination of A and B is important.
    – Mark Frazier
    Nov 29 '15 at 22:27












  • What is given means $AB-BA=lambda A+mu B$. Use that
    – Denis Düsseldorf
    Nov 29 '15 at 22:33










  • The linear combination of $A$ and $B$ is important because in general we do not know if $A$ and $B$ are diagonalizable w.r.t. the same basis. However, in the more general case that $AB-BA=0$, we do know that $A$ and $B$ are diagonalizable w.r.t. the same basis.
    – Hetebrij
    Nov 29 '15 at 23:41












  • In fact $A,B$ are simultaneously triangularizable. If you know Lie theory, then it is very easy. Otherwise do as follows. 1. Reduce the problem to the case $AB-BA=A$. 2. Calculate, by a recurrence reasoning, $A^kB-BA^k$. 3. Show that $A$ is nilpotent. 4. Consider $ker(A)$.
    – loup blanc
    Dec 3 '15 at 15:13


















  • Anything you tried to solve this problem?
    – Hetebrij
    Nov 29 '15 at 22:17










  • So, I know that eigenvectors are the solutions such that the determinant is zero when you know the eigenvalues to each. I assume that, since the matrices are distinct, their eigenvalues will have to be different for that matching eigenvector. Otherwise, I am very lost about why the linear combination of A and B is important.
    – Mark Frazier
    Nov 29 '15 at 22:27












  • What is given means $AB-BA=lambda A+mu B$. Use that
    – Denis Düsseldorf
    Nov 29 '15 at 22:33










  • The linear combination of $A$ and $B$ is important because in general we do not know if $A$ and $B$ are diagonalizable w.r.t. the same basis. However, in the more general case that $AB-BA=0$, we do know that $A$ and $B$ are diagonalizable w.r.t. the same basis.
    – Hetebrij
    Nov 29 '15 at 23:41












  • In fact $A,B$ are simultaneously triangularizable. If you know Lie theory, then it is very easy. Otherwise do as follows. 1. Reduce the problem to the case $AB-BA=A$. 2. Calculate, by a recurrence reasoning, $A^kB-BA^k$. 3. Show that $A$ is nilpotent. 4. Consider $ker(A)$.
    – loup blanc
    Dec 3 '15 at 15:13
















Anything you tried to solve this problem?
– Hetebrij
Nov 29 '15 at 22:17




Anything you tried to solve this problem?
– Hetebrij
Nov 29 '15 at 22:17












So, I know that eigenvectors are the solutions such that the determinant is zero when you know the eigenvalues to each. I assume that, since the matrices are distinct, their eigenvalues will have to be different for that matching eigenvector. Otherwise, I am very lost about why the linear combination of A and B is important.
– Mark Frazier
Nov 29 '15 at 22:27






So, I know that eigenvectors are the solutions such that the determinant is zero when you know the eigenvalues to each. I assume that, since the matrices are distinct, their eigenvalues will have to be different for that matching eigenvector. Otherwise, I am very lost about why the linear combination of A and B is important.
– Mark Frazier
Nov 29 '15 at 22:27














What is given means $AB-BA=lambda A+mu B$. Use that
– Denis Düsseldorf
Nov 29 '15 at 22:33




What is given means $AB-BA=lambda A+mu B$. Use that
– Denis Düsseldorf
Nov 29 '15 at 22:33












The linear combination of $A$ and $B$ is important because in general we do not know if $A$ and $B$ are diagonalizable w.r.t. the same basis. However, in the more general case that $AB-BA=0$, we do know that $A$ and $B$ are diagonalizable w.r.t. the same basis.
– Hetebrij
Nov 29 '15 at 23:41






The linear combination of $A$ and $B$ is important because in general we do not know if $A$ and $B$ are diagonalizable w.r.t. the same basis. However, in the more general case that $AB-BA=0$, we do know that $A$ and $B$ are diagonalizable w.r.t. the same basis.
– Hetebrij
Nov 29 '15 at 23:41














In fact $A,B$ are simultaneously triangularizable. If you know Lie theory, then it is very easy. Otherwise do as follows. 1. Reduce the problem to the case $AB-BA=A$. 2. Calculate, by a recurrence reasoning, $A^kB-BA^k$. 3. Show that $A$ is nilpotent. 4. Consider $ker(A)$.
– loup blanc
Dec 3 '15 at 15:13




In fact $A,B$ are simultaneously triangularizable. If you know Lie theory, then it is very easy. Otherwise do as follows. 1. Reduce the problem to the case $AB-BA=A$. 2. Calculate, by a recurrence reasoning, $A^kB-BA^k$. 3. Show that $A$ is nilpotent. 4. Consider $ker(A)$.
– loup blanc
Dec 3 '15 at 15:13










1 Answer
1






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oldest

votes


















1














It is assumed throughout that




The given matrices $A$ and $B$ are linearly independent in the $mathbb C$-vector space $M_n(mathbb C)$,

thus ignoring the unremarkable setup where one matrix is a multiple of the other.




My next guess is that the question doesn't focus upon a vanishing commutator of $A$ and $B$, hence consider
$$0:neq:[A,B]:=:AB-BA:=:alpha A+beta B$$
and at least one of the scalars $alpha,beta $ is not zero. Divide by a scalar, say $alphaneq0$, then set
$$B'=frac1alpha B;text{ and }; A'= A+beta B'quadtext{to obtain };
big[,A',B'big]:=:A',.$$

Thus in the sequel we may start w.l.o.g. from
$,mathbf{AB-BA=Aqquadqquad (star)}$



I decided to spill some more MathJax ink here & now, to give examples and a bit Lie algebra background as well. Who's not willing to go through all this should ...

simply jump: Goto The shared eigenvector !



Examples

how to satisfy the relation above:




  • If $,n=2$, then $,A=begin{pmatrix}0&1\ 0&0end{pmatrix}:$ and
    $;B=frac 12begin{pmatrix} -1&0\ 0&1end{pmatrix};$ do the job.
    $:begin{pmatrix}1\ 0end{pmatrix}$ is a joint eigenvector.


  • For $n=3$ one may consider
    $$A:=:begin{pmatrix}0&a_2&a_3\ 0&0&0\ 0&0&0end{pmatrix};,quad
    B:=:begin{pmatrix}b&0&0\ 0&b+1&0\ 0&0&b+1end{pmatrix}$$

    where $a_2,a_3,binmathbb C$,and $a_2,a_3$ not both zero.



It is not by accident that $A$ is represented by a nilpotent matrix since this cannot be avoided:

For each $ninmathbb N$ one has
$$big[,A^n,Bbig]:=:n,A^n,$$
the proof by induction using
$,[AA^{n-1},B] = A[A^{n-1},B] + [A,B],A^{n-1}$ (derivation property) is straightforward. Let $|cdot|$ be a submultiplicative matrix norm, then
$$n|A^n|:=:|A^nB-BA^n|;leq;2|B| |A^n|quadforall,n$$
which implies $A^n=0,$ at the latest when $,n>2|B|$. Thus $A$ is nilpotent.



The shared eigenvector

for $B$ and $A$ does exist: Pick an eigenvector $wneq 0$ to an eigenvalue $mu$ of $B$ such that $,mu-1,$ is no eigenvalue of $B$. Insert $w$ into $(star)$
$$BAw:=:ABw - Aw:=:(mu-1)AwquadLongrightarrow; Aw:=:0,,$$
thus $w$ is also an eigenvector of $A$, to the eigenvalue zero.



Lie algebra background
$A$ and $B$ span a complex Lie algebra with Lie bracket given by $(star)$. It is up to (Lie algebra) isomorphism the unique complex Lie algebra, which is 2-dimensional and nonabelian. It is solvable, but not nilpotent.

Lie's theorem guarantees (in any matrix representation) the existence of a shared eigenvector for all elements of a solvable Lie algebra. One implication is that all matrices of a representation may be chosen to be triangular.



Related posts




  • $AB + BA = A implies A$ and $B$ have a common eigenvector


  • Questions about non-abelian 2 dimensional lie algebras


  • A Problem about Common Eigenvector


  • Representations of the two dimensional non-abelian Lie algebra @MO


  • Examples of finite dimensional non simple non abelian Lie algebras @MO







share|cite|improve this answer























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    1 Answer
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    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    1














    It is assumed throughout that




    The given matrices $A$ and $B$ are linearly independent in the $mathbb C$-vector space $M_n(mathbb C)$,

    thus ignoring the unremarkable setup where one matrix is a multiple of the other.




    My next guess is that the question doesn't focus upon a vanishing commutator of $A$ and $B$, hence consider
    $$0:neq:[A,B]:=:AB-BA:=:alpha A+beta B$$
    and at least one of the scalars $alpha,beta $ is not zero. Divide by a scalar, say $alphaneq0$, then set
    $$B'=frac1alpha B;text{ and }; A'= A+beta B'quadtext{to obtain };
    big[,A',B'big]:=:A',.$$

    Thus in the sequel we may start w.l.o.g. from
    $,mathbf{AB-BA=Aqquadqquad (star)}$



    I decided to spill some more MathJax ink here & now, to give examples and a bit Lie algebra background as well. Who's not willing to go through all this should ...

    simply jump: Goto The shared eigenvector !



    Examples

    how to satisfy the relation above:




    • If $,n=2$, then $,A=begin{pmatrix}0&1\ 0&0end{pmatrix}:$ and
      $;B=frac 12begin{pmatrix} -1&0\ 0&1end{pmatrix};$ do the job.
      $:begin{pmatrix}1\ 0end{pmatrix}$ is a joint eigenvector.


    • For $n=3$ one may consider
      $$A:=:begin{pmatrix}0&a_2&a_3\ 0&0&0\ 0&0&0end{pmatrix};,quad
      B:=:begin{pmatrix}b&0&0\ 0&b+1&0\ 0&0&b+1end{pmatrix}$$

      where $a_2,a_3,binmathbb C$,and $a_2,a_3$ not both zero.



    It is not by accident that $A$ is represented by a nilpotent matrix since this cannot be avoided:

    For each $ninmathbb N$ one has
    $$big[,A^n,Bbig]:=:n,A^n,$$
    the proof by induction using
    $,[AA^{n-1},B] = A[A^{n-1},B] + [A,B],A^{n-1}$ (derivation property) is straightforward. Let $|cdot|$ be a submultiplicative matrix norm, then
    $$n|A^n|:=:|A^nB-BA^n|;leq;2|B| |A^n|quadforall,n$$
    which implies $A^n=0,$ at the latest when $,n>2|B|$. Thus $A$ is nilpotent.



    The shared eigenvector

    for $B$ and $A$ does exist: Pick an eigenvector $wneq 0$ to an eigenvalue $mu$ of $B$ such that $,mu-1,$ is no eigenvalue of $B$. Insert $w$ into $(star)$
    $$BAw:=:ABw - Aw:=:(mu-1)AwquadLongrightarrow; Aw:=:0,,$$
    thus $w$ is also an eigenvector of $A$, to the eigenvalue zero.



    Lie algebra background
    $A$ and $B$ span a complex Lie algebra with Lie bracket given by $(star)$. It is up to (Lie algebra) isomorphism the unique complex Lie algebra, which is 2-dimensional and nonabelian. It is solvable, but not nilpotent.

    Lie's theorem guarantees (in any matrix representation) the existence of a shared eigenvector for all elements of a solvable Lie algebra. One implication is that all matrices of a representation may be chosen to be triangular.



    Related posts




    • $AB + BA = A implies A$ and $B$ have a common eigenvector


    • Questions about non-abelian 2 dimensional lie algebras


    • A Problem about Common Eigenvector


    • Representations of the two dimensional non-abelian Lie algebra @MO


    • Examples of finite dimensional non simple non abelian Lie algebras @MO







    share|cite|improve this answer




























      1














      It is assumed throughout that




      The given matrices $A$ and $B$ are linearly independent in the $mathbb C$-vector space $M_n(mathbb C)$,

      thus ignoring the unremarkable setup where one matrix is a multiple of the other.




      My next guess is that the question doesn't focus upon a vanishing commutator of $A$ and $B$, hence consider
      $$0:neq:[A,B]:=:AB-BA:=:alpha A+beta B$$
      and at least one of the scalars $alpha,beta $ is not zero. Divide by a scalar, say $alphaneq0$, then set
      $$B'=frac1alpha B;text{ and }; A'= A+beta B'quadtext{to obtain };
      big[,A',B'big]:=:A',.$$

      Thus in the sequel we may start w.l.o.g. from
      $,mathbf{AB-BA=Aqquadqquad (star)}$



      I decided to spill some more MathJax ink here & now, to give examples and a bit Lie algebra background as well. Who's not willing to go through all this should ...

      simply jump: Goto The shared eigenvector !



      Examples

      how to satisfy the relation above:




      • If $,n=2$, then $,A=begin{pmatrix}0&1\ 0&0end{pmatrix}:$ and
        $;B=frac 12begin{pmatrix} -1&0\ 0&1end{pmatrix};$ do the job.
        $:begin{pmatrix}1\ 0end{pmatrix}$ is a joint eigenvector.


      • For $n=3$ one may consider
        $$A:=:begin{pmatrix}0&a_2&a_3\ 0&0&0\ 0&0&0end{pmatrix};,quad
        B:=:begin{pmatrix}b&0&0\ 0&b+1&0\ 0&0&b+1end{pmatrix}$$

        where $a_2,a_3,binmathbb C$,and $a_2,a_3$ not both zero.



      It is not by accident that $A$ is represented by a nilpotent matrix since this cannot be avoided:

      For each $ninmathbb N$ one has
      $$big[,A^n,Bbig]:=:n,A^n,$$
      the proof by induction using
      $,[AA^{n-1},B] = A[A^{n-1},B] + [A,B],A^{n-1}$ (derivation property) is straightforward. Let $|cdot|$ be a submultiplicative matrix norm, then
      $$n|A^n|:=:|A^nB-BA^n|;leq;2|B| |A^n|quadforall,n$$
      which implies $A^n=0,$ at the latest when $,n>2|B|$. Thus $A$ is nilpotent.



      The shared eigenvector

      for $B$ and $A$ does exist: Pick an eigenvector $wneq 0$ to an eigenvalue $mu$ of $B$ such that $,mu-1,$ is no eigenvalue of $B$. Insert $w$ into $(star)$
      $$BAw:=:ABw - Aw:=:(mu-1)AwquadLongrightarrow; Aw:=:0,,$$
      thus $w$ is also an eigenvector of $A$, to the eigenvalue zero.



      Lie algebra background
      $A$ and $B$ span a complex Lie algebra with Lie bracket given by $(star)$. It is up to (Lie algebra) isomorphism the unique complex Lie algebra, which is 2-dimensional and nonabelian. It is solvable, but not nilpotent.

      Lie's theorem guarantees (in any matrix representation) the existence of a shared eigenvector for all elements of a solvable Lie algebra. One implication is that all matrices of a representation may be chosen to be triangular.



      Related posts




      • $AB + BA = A implies A$ and $B$ have a common eigenvector


      • Questions about non-abelian 2 dimensional lie algebras


      • A Problem about Common Eigenvector


      • Representations of the two dimensional non-abelian Lie algebra @MO


      • Examples of finite dimensional non simple non abelian Lie algebras @MO







      share|cite|improve this answer


























        1












        1








        1






        It is assumed throughout that




        The given matrices $A$ and $B$ are linearly independent in the $mathbb C$-vector space $M_n(mathbb C)$,

        thus ignoring the unremarkable setup where one matrix is a multiple of the other.




        My next guess is that the question doesn't focus upon a vanishing commutator of $A$ and $B$, hence consider
        $$0:neq:[A,B]:=:AB-BA:=:alpha A+beta B$$
        and at least one of the scalars $alpha,beta $ is not zero. Divide by a scalar, say $alphaneq0$, then set
        $$B'=frac1alpha B;text{ and }; A'= A+beta B'quadtext{to obtain };
        big[,A',B'big]:=:A',.$$

        Thus in the sequel we may start w.l.o.g. from
        $,mathbf{AB-BA=Aqquadqquad (star)}$



        I decided to spill some more MathJax ink here & now, to give examples and a bit Lie algebra background as well. Who's not willing to go through all this should ...

        simply jump: Goto The shared eigenvector !



        Examples

        how to satisfy the relation above:




        • If $,n=2$, then $,A=begin{pmatrix}0&1\ 0&0end{pmatrix}:$ and
          $;B=frac 12begin{pmatrix} -1&0\ 0&1end{pmatrix};$ do the job.
          $:begin{pmatrix}1\ 0end{pmatrix}$ is a joint eigenvector.


        • For $n=3$ one may consider
          $$A:=:begin{pmatrix}0&a_2&a_3\ 0&0&0\ 0&0&0end{pmatrix};,quad
          B:=:begin{pmatrix}b&0&0\ 0&b+1&0\ 0&0&b+1end{pmatrix}$$

          where $a_2,a_3,binmathbb C$,and $a_2,a_3$ not both zero.



        It is not by accident that $A$ is represented by a nilpotent matrix since this cannot be avoided:

        For each $ninmathbb N$ one has
        $$big[,A^n,Bbig]:=:n,A^n,$$
        the proof by induction using
        $,[AA^{n-1},B] = A[A^{n-1},B] + [A,B],A^{n-1}$ (derivation property) is straightforward. Let $|cdot|$ be a submultiplicative matrix norm, then
        $$n|A^n|:=:|A^nB-BA^n|;leq;2|B| |A^n|quadforall,n$$
        which implies $A^n=0,$ at the latest when $,n>2|B|$. Thus $A$ is nilpotent.



        The shared eigenvector

        for $B$ and $A$ does exist: Pick an eigenvector $wneq 0$ to an eigenvalue $mu$ of $B$ such that $,mu-1,$ is no eigenvalue of $B$. Insert $w$ into $(star)$
        $$BAw:=:ABw - Aw:=:(mu-1)AwquadLongrightarrow; Aw:=:0,,$$
        thus $w$ is also an eigenvector of $A$, to the eigenvalue zero.



        Lie algebra background
        $A$ and $B$ span a complex Lie algebra with Lie bracket given by $(star)$. It is up to (Lie algebra) isomorphism the unique complex Lie algebra, which is 2-dimensional and nonabelian. It is solvable, but not nilpotent.

        Lie's theorem guarantees (in any matrix representation) the existence of a shared eigenvector for all elements of a solvable Lie algebra. One implication is that all matrices of a representation may be chosen to be triangular.



        Related posts




        • $AB + BA = A implies A$ and $B$ have a common eigenvector


        • Questions about non-abelian 2 dimensional lie algebras


        • A Problem about Common Eigenvector


        • Representations of the two dimensional non-abelian Lie algebra @MO


        • Examples of finite dimensional non simple non abelian Lie algebras @MO







        share|cite|improve this answer














        It is assumed throughout that




        The given matrices $A$ and $B$ are linearly independent in the $mathbb C$-vector space $M_n(mathbb C)$,

        thus ignoring the unremarkable setup where one matrix is a multiple of the other.




        My next guess is that the question doesn't focus upon a vanishing commutator of $A$ and $B$, hence consider
        $$0:neq:[A,B]:=:AB-BA:=:alpha A+beta B$$
        and at least one of the scalars $alpha,beta $ is not zero. Divide by a scalar, say $alphaneq0$, then set
        $$B'=frac1alpha B;text{ and }; A'= A+beta B'quadtext{to obtain };
        big[,A',B'big]:=:A',.$$

        Thus in the sequel we may start w.l.o.g. from
        $,mathbf{AB-BA=Aqquadqquad (star)}$



        I decided to spill some more MathJax ink here & now, to give examples and a bit Lie algebra background as well. Who's not willing to go through all this should ...

        simply jump: Goto The shared eigenvector !



        Examples

        how to satisfy the relation above:




        • If $,n=2$, then $,A=begin{pmatrix}0&1\ 0&0end{pmatrix}:$ and
          $;B=frac 12begin{pmatrix} -1&0\ 0&1end{pmatrix};$ do the job.
          $:begin{pmatrix}1\ 0end{pmatrix}$ is a joint eigenvector.


        • For $n=3$ one may consider
          $$A:=:begin{pmatrix}0&a_2&a_3\ 0&0&0\ 0&0&0end{pmatrix};,quad
          B:=:begin{pmatrix}b&0&0\ 0&b+1&0\ 0&0&b+1end{pmatrix}$$

          where $a_2,a_3,binmathbb C$,and $a_2,a_3$ not both zero.



        It is not by accident that $A$ is represented by a nilpotent matrix since this cannot be avoided:

        For each $ninmathbb N$ one has
        $$big[,A^n,Bbig]:=:n,A^n,$$
        the proof by induction using
        $,[AA^{n-1},B] = A[A^{n-1},B] + [A,B],A^{n-1}$ (derivation property) is straightforward. Let $|cdot|$ be a submultiplicative matrix norm, then
        $$n|A^n|:=:|A^nB-BA^n|;leq;2|B| |A^n|quadforall,n$$
        which implies $A^n=0,$ at the latest when $,n>2|B|$. Thus $A$ is nilpotent.



        The shared eigenvector

        for $B$ and $A$ does exist: Pick an eigenvector $wneq 0$ to an eigenvalue $mu$ of $B$ such that $,mu-1,$ is no eigenvalue of $B$. Insert $w$ into $(star)$
        $$BAw:=:ABw - Aw:=:(mu-1)AwquadLongrightarrow; Aw:=:0,,$$
        thus $w$ is also an eigenvector of $A$, to the eigenvalue zero.



        Lie algebra background
        $A$ and $B$ span a complex Lie algebra with Lie bracket given by $(star)$. It is up to (Lie algebra) isomorphism the unique complex Lie algebra, which is 2-dimensional and nonabelian. It is solvable, but not nilpotent.

        Lie's theorem guarantees (in any matrix representation) the existence of a shared eigenvector for all elements of a solvable Lie algebra. One implication is that all matrices of a representation may be chosen to be triangular.



        Related posts




        • $AB + BA = A implies A$ and $B$ have a common eigenvector


        • Questions about non-abelian 2 dimensional lie algebras


        • A Problem about Common Eigenvector


        • Representations of the two dimensional non-abelian Lie algebra @MO


        • Examples of finite dimensional non simple non abelian Lie algebras @MO








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        edited yesterday

























        answered Jan 4 at 22:22









        HannoHanno

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