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quadratic equations on 2 by 2 matrices


Differences between Real Matrices and Complex matrices.Systems of equations with matricesHow to determine if a set is a subspace of the vector space of all complex $2times 2$ matrices?Counting Distinct matricesOn multiplying symmetric matrices by diagonal matrices with roots of unityFind Matrix C relating two symmetric matrices A and B (Such that AC = B)Structure-preserving matricesIs there a case that 2 positive matrices multiplication returning 0 element?






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margin-bottom:0;

.everyonelovesstackoverflowposition:absolute;height:1px;width:1px;opacity:0;top:0;left:0;pointer-events:none;








3












$begingroup$


For real non-zero $2times 2$ matrices, can we say:




For any $A,B$, there are at most two matrices $X$ such that $XX + AX + B =0$




Is there a way to see this without going in the direction of writing the open form result of $XX + AX + B$ and solve each element is equal to zero, etc.?










share|cite|improve this question











$endgroup$




















    3












    $begingroup$


    For real non-zero $2times 2$ matrices, can we say:




    For any $A,B$, there are at most two matrices $X$ such that $XX + AX + B =0$




    Is there a way to see this without going in the direction of writing the open form result of $XX + AX + B$ and solve each element is equal to zero, etc.?










    share|cite|improve this question











    $endgroup$
















      3












      3








      3





      $begingroup$


      For real non-zero $2times 2$ matrices, can we say:




      For any $A,B$, there are at most two matrices $X$ such that $XX + AX + B =0$




      Is there a way to see this without going in the direction of writing the open form result of $XX + AX + B$ and solve each element is equal to zero, etc.?










      share|cite|improve this question











      $endgroup$




      For real non-zero $2times 2$ matrices, can we say:




      For any $A,B$, there are at most two matrices $X$ such that $XX + AX + B =0$




      Is there a way to see this without going in the direction of writing the open form result of $XX + AX + B$ and solve each element is equal to zero, etc.?







      linear-algebra matrices quadratic-forms






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 8 hours ago







      independentvariable

















      asked 8 hours ago









      independentvariableindependentvariable

      3631 silver badge11 bronze badges




      3631 silver badge11 bronze badges























          2 Answers
          2






          active

          oldest

          votes


















          4














          $begingroup$

          No. E.g. over $mathbb R$, we have $pmatrix1&t\ 0&-1^2=I$ for any $t$. It also follows that
          $$
          pmatrix1&t\ 0&-1^2+pmatrix0&0\ 0&1pmatrix1&t\ 0&-1+pmatrix-1&0\ 0&0=0
          $$

          for any $t$.



          Edit. Another counterexample (inspired by Dietrich Burde's answer). Let $u,v$ be any two vectors such that $v^Tu=0$. Then $X=uv^T$ is a solution to the equation $X^2=0$. In particular, if the underlying field is $GF(2)$ (the field consisting of only two elements $0$ and $1$, with $1+1=0$), the equation $Y^2=0$ for a $2times2$ matrix $Y$ has exactly four solutions
          $$
          Yinleftpmatrix0&0\ 0&0, pmatrix0&1\ 0&0, pmatrix0&0\ 1&0, pmatrix1&1\ 1&1right.
          $$

          And so does the equation $X^2=I$ over $GF(2)$: just put $X=Y+I$.






          share|cite|improve this answer











          $endgroup$














          • $begingroup$
            how did you come up with this so fast? may I ask your technique?
            $endgroup$
            – independentvariable
            8 hours ago






          • 1




            $begingroup$
            @independentvariable It may not be hard for someone to figure out a counterexample, but I just happen to know that $X^2=I$ has infinitely many solutions over a field of characteristic zero.
            $endgroup$
            – user1551
            7 hours ago


















          3














          $begingroup$

          No, take for example $A=0$ and
          $$
          B=beginpmatrix 0 & 1 cr 0 & 0 endpmatrix.
          $$

          Then $X^2=-B$ has no solution, since $X$ is nilpotent, so that $X^2=0$ but $Bneq 0$.






          share|cite|improve this answer









          $endgroup$














          • $begingroup$
            The matrices are non zero though.
            $endgroup$
            – independentvariable
            8 hours ago







          • 1




            $begingroup$
            It doesn't matter really. You can adapt this example with nonzero $A$. Quadratic matrix solutions need not have any solution.
            $endgroup$
            – Dietrich Burde
            7 hours ago











          • $begingroup$
            But the question is about showing it can have more than 2, isnt it?
            $endgroup$
            – independentvariable
            7 hours ago






          • 1




            $begingroup$
            Yes, sorry, I thought of "at least". So quadratic matrix equations can have no solution, or a number of solutions including infinity. It comes from the fact that the matrix algebra is not commutative.
            $endgroup$
            – Dietrich Burde
            7 hours ago











          • $begingroup$
            Still, good for knowledge :)
            $endgroup$
            – independentvariable
            7 hours ago












          Your Answer








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          2 Answers
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          2 Answers
          2






          active

          oldest

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          active

          oldest

          votes






          active

          oldest

          votes









          4














          $begingroup$

          No. E.g. over $mathbb R$, we have $pmatrix1&t\ 0&-1^2=I$ for any $t$. It also follows that
          $$
          pmatrix1&t\ 0&-1^2+pmatrix0&0\ 0&1pmatrix1&t\ 0&-1+pmatrix-1&0\ 0&0=0
          $$

          for any $t$.



          Edit. Another counterexample (inspired by Dietrich Burde's answer). Let $u,v$ be any two vectors such that $v^Tu=0$. Then $X=uv^T$ is a solution to the equation $X^2=0$. In particular, if the underlying field is $GF(2)$ (the field consisting of only two elements $0$ and $1$, with $1+1=0$), the equation $Y^2=0$ for a $2times2$ matrix $Y$ has exactly four solutions
          $$
          Yinleftpmatrix0&0\ 0&0, pmatrix0&1\ 0&0, pmatrix0&0\ 1&0, pmatrix1&1\ 1&1right.
          $$

          And so does the equation $X^2=I$ over $GF(2)$: just put $X=Y+I$.






          share|cite|improve this answer











          $endgroup$














          • $begingroup$
            how did you come up with this so fast? may I ask your technique?
            $endgroup$
            – independentvariable
            8 hours ago






          • 1




            $begingroup$
            @independentvariable It may not be hard for someone to figure out a counterexample, but I just happen to know that $X^2=I$ has infinitely many solutions over a field of characteristic zero.
            $endgroup$
            – user1551
            7 hours ago















          4














          $begingroup$

          No. E.g. over $mathbb R$, we have $pmatrix1&t\ 0&-1^2=I$ for any $t$. It also follows that
          $$
          pmatrix1&t\ 0&-1^2+pmatrix0&0\ 0&1pmatrix1&t\ 0&-1+pmatrix-1&0\ 0&0=0
          $$

          for any $t$.



          Edit. Another counterexample (inspired by Dietrich Burde's answer). Let $u,v$ be any two vectors such that $v^Tu=0$. Then $X=uv^T$ is a solution to the equation $X^2=0$. In particular, if the underlying field is $GF(2)$ (the field consisting of only two elements $0$ and $1$, with $1+1=0$), the equation $Y^2=0$ for a $2times2$ matrix $Y$ has exactly four solutions
          $$
          Yinleftpmatrix0&0\ 0&0, pmatrix0&1\ 0&0, pmatrix0&0\ 1&0, pmatrix1&1\ 1&1right.
          $$

          And so does the equation $X^2=I$ over $GF(2)$: just put $X=Y+I$.






          share|cite|improve this answer











          $endgroup$














          • $begingroup$
            how did you come up with this so fast? may I ask your technique?
            $endgroup$
            – independentvariable
            8 hours ago






          • 1




            $begingroup$
            @independentvariable It may not be hard for someone to figure out a counterexample, but I just happen to know that $X^2=I$ has infinitely many solutions over a field of characteristic zero.
            $endgroup$
            – user1551
            7 hours ago













          4














          4










          4







          $begingroup$

          No. E.g. over $mathbb R$, we have $pmatrix1&t\ 0&-1^2=I$ for any $t$. It also follows that
          $$
          pmatrix1&t\ 0&-1^2+pmatrix0&0\ 0&1pmatrix1&t\ 0&-1+pmatrix-1&0\ 0&0=0
          $$

          for any $t$.



          Edit. Another counterexample (inspired by Dietrich Burde's answer). Let $u,v$ be any two vectors such that $v^Tu=0$. Then $X=uv^T$ is a solution to the equation $X^2=0$. In particular, if the underlying field is $GF(2)$ (the field consisting of only two elements $0$ and $1$, with $1+1=0$), the equation $Y^2=0$ for a $2times2$ matrix $Y$ has exactly four solutions
          $$
          Yinleftpmatrix0&0\ 0&0, pmatrix0&1\ 0&0, pmatrix0&0\ 1&0, pmatrix1&1\ 1&1right.
          $$

          And so does the equation $X^2=I$ over $GF(2)$: just put $X=Y+I$.






          share|cite|improve this answer











          $endgroup$



          No. E.g. over $mathbb R$, we have $pmatrix1&t\ 0&-1^2=I$ for any $t$. It also follows that
          $$
          pmatrix1&t\ 0&-1^2+pmatrix0&0\ 0&1pmatrix1&t\ 0&-1+pmatrix-1&0\ 0&0=0
          $$

          for any $t$.



          Edit. Another counterexample (inspired by Dietrich Burde's answer). Let $u,v$ be any two vectors such that $v^Tu=0$. Then $X=uv^T$ is a solution to the equation $X^2=0$. In particular, if the underlying field is $GF(2)$ (the field consisting of only two elements $0$ and $1$, with $1+1=0$), the equation $Y^2=0$ for a $2times2$ matrix $Y$ has exactly four solutions
          $$
          Yinleftpmatrix0&0\ 0&0, pmatrix0&1\ 0&0, pmatrix0&0\ 1&0, pmatrix1&1\ 1&1right.
          $$

          And so does the equation $X^2=I$ over $GF(2)$: just put $X=Y+I$.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited 7 hours ago

























          answered 8 hours ago









          user1551user1551

          81.1k6 gold badges74 silver badges137 bronze badges




          81.1k6 gold badges74 silver badges137 bronze badges














          • $begingroup$
            how did you come up with this so fast? may I ask your technique?
            $endgroup$
            – independentvariable
            8 hours ago






          • 1




            $begingroup$
            @independentvariable It may not be hard for someone to figure out a counterexample, but I just happen to know that $X^2=I$ has infinitely many solutions over a field of characteristic zero.
            $endgroup$
            – user1551
            7 hours ago
















          • $begingroup$
            how did you come up with this so fast? may I ask your technique?
            $endgroup$
            – independentvariable
            8 hours ago






          • 1




            $begingroup$
            @independentvariable It may not be hard for someone to figure out a counterexample, but I just happen to know that $X^2=I$ has infinitely many solutions over a field of characteristic zero.
            $endgroup$
            – user1551
            7 hours ago















          $begingroup$
          how did you come up with this so fast? may I ask your technique?
          $endgroup$
          – independentvariable
          8 hours ago




          $begingroup$
          how did you come up with this so fast? may I ask your technique?
          $endgroup$
          – independentvariable
          8 hours ago




          1




          1




          $begingroup$
          @independentvariable It may not be hard for someone to figure out a counterexample, but I just happen to know that $X^2=I$ has infinitely many solutions over a field of characteristic zero.
          $endgroup$
          – user1551
          7 hours ago




          $begingroup$
          @independentvariable It may not be hard for someone to figure out a counterexample, but I just happen to know that $X^2=I$ has infinitely many solutions over a field of characteristic zero.
          $endgroup$
          – user1551
          7 hours ago













          3














          $begingroup$

          No, take for example $A=0$ and
          $$
          B=beginpmatrix 0 & 1 cr 0 & 0 endpmatrix.
          $$

          Then $X^2=-B$ has no solution, since $X$ is nilpotent, so that $X^2=0$ but $Bneq 0$.






          share|cite|improve this answer









          $endgroup$














          • $begingroup$
            The matrices are non zero though.
            $endgroup$
            – independentvariable
            8 hours ago







          • 1




            $begingroup$
            It doesn't matter really. You can adapt this example with nonzero $A$. Quadratic matrix solutions need not have any solution.
            $endgroup$
            – Dietrich Burde
            7 hours ago











          • $begingroup$
            But the question is about showing it can have more than 2, isnt it?
            $endgroup$
            – independentvariable
            7 hours ago






          • 1




            $begingroup$
            Yes, sorry, I thought of "at least". So quadratic matrix equations can have no solution, or a number of solutions including infinity. It comes from the fact that the matrix algebra is not commutative.
            $endgroup$
            – Dietrich Burde
            7 hours ago











          • $begingroup$
            Still, good for knowledge :)
            $endgroup$
            – independentvariable
            7 hours ago















          3














          $begingroup$

          No, take for example $A=0$ and
          $$
          B=beginpmatrix 0 & 1 cr 0 & 0 endpmatrix.
          $$

          Then $X^2=-B$ has no solution, since $X$ is nilpotent, so that $X^2=0$ but $Bneq 0$.






          share|cite|improve this answer









          $endgroup$














          • $begingroup$
            The matrices are non zero though.
            $endgroup$
            – independentvariable
            8 hours ago







          • 1




            $begingroup$
            It doesn't matter really. You can adapt this example with nonzero $A$. Quadratic matrix solutions need not have any solution.
            $endgroup$
            – Dietrich Burde
            7 hours ago











          • $begingroup$
            But the question is about showing it can have more than 2, isnt it?
            $endgroup$
            – independentvariable
            7 hours ago






          • 1




            $begingroup$
            Yes, sorry, I thought of "at least". So quadratic matrix equations can have no solution, or a number of solutions including infinity. It comes from the fact that the matrix algebra is not commutative.
            $endgroup$
            – Dietrich Burde
            7 hours ago











          • $begingroup$
            Still, good for knowledge :)
            $endgroup$
            – independentvariable
            7 hours ago













          3














          3










          3







          $begingroup$

          No, take for example $A=0$ and
          $$
          B=beginpmatrix 0 & 1 cr 0 & 0 endpmatrix.
          $$

          Then $X^2=-B$ has no solution, since $X$ is nilpotent, so that $X^2=0$ but $Bneq 0$.






          share|cite|improve this answer









          $endgroup$



          No, take for example $A=0$ and
          $$
          B=beginpmatrix 0 & 1 cr 0 & 0 endpmatrix.
          $$

          Then $X^2=-B$ has no solution, since $X$ is nilpotent, so that $X^2=0$ but $Bneq 0$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 8 hours ago









          Dietrich BurdeDietrich Burde

          87.9k6 gold badges50 silver badges111 bronze badges




          87.9k6 gold badges50 silver badges111 bronze badges














          • $begingroup$
            The matrices are non zero though.
            $endgroup$
            – independentvariable
            8 hours ago







          • 1




            $begingroup$
            It doesn't matter really. You can adapt this example with nonzero $A$. Quadratic matrix solutions need not have any solution.
            $endgroup$
            – Dietrich Burde
            7 hours ago











          • $begingroup$
            But the question is about showing it can have more than 2, isnt it?
            $endgroup$
            – independentvariable
            7 hours ago






          • 1




            $begingroup$
            Yes, sorry, I thought of "at least". So quadratic matrix equations can have no solution, or a number of solutions including infinity. It comes from the fact that the matrix algebra is not commutative.
            $endgroup$
            – Dietrich Burde
            7 hours ago











          • $begingroup$
            Still, good for knowledge :)
            $endgroup$
            – independentvariable
            7 hours ago
















          • $begingroup$
            The matrices are non zero though.
            $endgroup$
            – independentvariable
            8 hours ago







          • 1




            $begingroup$
            It doesn't matter really. You can adapt this example with nonzero $A$. Quadratic matrix solutions need not have any solution.
            $endgroup$
            – Dietrich Burde
            7 hours ago











          • $begingroup$
            But the question is about showing it can have more than 2, isnt it?
            $endgroup$
            – independentvariable
            7 hours ago






          • 1




            $begingroup$
            Yes, sorry, I thought of "at least". So quadratic matrix equations can have no solution, or a number of solutions including infinity. It comes from the fact that the matrix algebra is not commutative.
            $endgroup$
            – Dietrich Burde
            7 hours ago











          • $begingroup$
            Still, good for knowledge :)
            $endgroup$
            – independentvariable
            7 hours ago















          $begingroup$
          The matrices are non zero though.
          $endgroup$
          – independentvariable
          8 hours ago





          $begingroup$
          The matrices are non zero though.
          $endgroup$
          – independentvariable
          8 hours ago





          1




          1




          $begingroup$
          It doesn't matter really. You can adapt this example with nonzero $A$. Quadratic matrix solutions need not have any solution.
          $endgroup$
          – Dietrich Burde
          7 hours ago





          $begingroup$
          It doesn't matter really. You can adapt this example with nonzero $A$. Quadratic matrix solutions need not have any solution.
          $endgroup$
          – Dietrich Burde
          7 hours ago













          $begingroup$
          But the question is about showing it can have more than 2, isnt it?
          $endgroup$
          – independentvariable
          7 hours ago




          $begingroup$
          But the question is about showing it can have more than 2, isnt it?
          $endgroup$
          – independentvariable
          7 hours ago




          1




          1




          $begingroup$
          Yes, sorry, I thought of "at least". So quadratic matrix equations can have no solution, or a number of solutions including infinity. It comes from the fact that the matrix algebra is not commutative.
          $endgroup$
          – Dietrich Burde
          7 hours ago





          $begingroup$
          Yes, sorry, I thought of "at least". So quadratic matrix equations can have no solution, or a number of solutions including infinity. It comes from the fact that the matrix algebra is not commutative.
          $endgroup$
          – Dietrich Burde
          7 hours ago













          $begingroup$
          Still, good for knowledge :)
          $endgroup$
          – independentvariable
          7 hours ago




          $begingroup$
          Still, good for knowledge :)
          $endgroup$
          – independentvariable
          7 hours ago


















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