prove that $A$ is diagonalizable if $A^3-3A^2-A+3I_n = 0$ [on hold]Hermitian Matrices are DiagonalizableA question about diagonalizable.Show that matrix $A$ is NOT diagonalizable.Prove that A … is not diagonalizableProve a matrix is not diagonalizableHow to use inner products in C(n) to prove normal matrix is unitarily diagonalizable after knowing that normal matrix is diagonalizable?Prove that $A$ is diagonalizable.Prove that a general matrix is diagonalizableDetermine $a$ to make matrix $A$ diagonalizableDiagonalizable block-diagonal matrix

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prove that $A$ is diagonalizable if $A^3-3A^2-A+3I_n = 0$ [on hold]


Hermitian Matrices are DiagonalizableA question about diagonalizable.Show that matrix $A$ is NOT diagonalizable.Prove that A … is not diagonalizableProve a matrix is not diagonalizableHow to use inner products in C(n) to prove normal matrix is unitarily diagonalizable after knowing that normal matrix is diagonalizable?Prove that $A$ is diagonalizable.Prove that a general matrix is diagonalizableDetermine $a$ to make matrix $A$ diagonalizableDiagonalizable block-diagonal matrix













-1












$begingroup$


We have :



$A^3-3A^2-A+3I_n = 0 $



how can i prove that A is diagonalizable .



I don't know how to do when A is written this way










share|cite|improve this question











$endgroup$



put on hold as off-topic by user21820, RRL, Saad, Cesareo, mrtaurho 2 days ago


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – user21820, RRL, Saad, Cesareo, mrtaurho
If this question can be reworded to fit the rules in the help center, please edit the question.











  • 2




    $begingroup$
    Note that if $A=operatornameId_n$, then $A^3-3A^2-A+3operatornameId_n=0$, in spite of the fact that the only root of the characteristic polynomial of $operatornameId_n$ has multiplicity $n$. So, no, you don't have to prove that all roots of the characteristic polynomial of $A$ have multiplicity $1$.
    $endgroup$
    – José Carlos Santos
    Apr 4 at 20:37










  • $begingroup$
    yes , you're right i edit my mistake
    $endgroup$
    – JoshuaK
    Apr 4 at 20:39










  • $begingroup$
    What does "when $A$ is written this way" mean?
    $endgroup$
    – anomaly
    Apr 5 at 0:42















-1












$begingroup$


We have :



$A^3-3A^2-A+3I_n = 0 $



how can i prove that A is diagonalizable .



I don't know how to do when A is written this way










share|cite|improve this question











$endgroup$



put on hold as off-topic by user21820, RRL, Saad, Cesareo, mrtaurho 2 days ago


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – user21820, RRL, Saad, Cesareo, mrtaurho
If this question can be reworded to fit the rules in the help center, please edit the question.











  • 2




    $begingroup$
    Note that if $A=operatornameId_n$, then $A^3-3A^2-A+3operatornameId_n=0$, in spite of the fact that the only root of the characteristic polynomial of $operatornameId_n$ has multiplicity $n$. So, no, you don't have to prove that all roots of the characteristic polynomial of $A$ have multiplicity $1$.
    $endgroup$
    – José Carlos Santos
    Apr 4 at 20:37










  • $begingroup$
    yes , you're right i edit my mistake
    $endgroup$
    – JoshuaK
    Apr 4 at 20:39










  • $begingroup$
    What does "when $A$ is written this way" mean?
    $endgroup$
    – anomaly
    Apr 5 at 0:42













-1












-1








-1





$begingroup$


We have :



$A^3-3A^2-A+3I_n = 0 $



how can i prove that A is diagonalizable .



I don't know how to do when A is written this way










share|cite|improve this question











$endgroup$




We have :



$A^3-3A^2-A+3I_n = 0 $



how can i prove that A is diagonalizable .



I don't know how to do when A is written this way







linear-algebra matrices eigenvalues-eigenvectors diagonalization






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Apr 5 at 2:23









user21820

40.1k544162




40.1k544162










asked Apr 4 at 20:34









JoshuaKJoshuaK

305




305




put on hold as off-topic by user21820, RRL, Saad, Cesareo, mrtaurho 2 days ago


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – user21820, RRL, Saad, Cesareo, mrtaurho
If this question can be reworded to fit the rules in the help center, please edit the question.







put on hold as off-topic by user21820, RRL, Saad, Cesareo, mrtaurho 2 days ago


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – user21820, RRL, Saad, Cesareo, mrtaurho
If this question can be reworded to fit the rules in the help center, please edit the question.







  • 2




    $begingroup$
    Note that if $A=operatornameId_n$, then $A^3-3A^2-A+3operatornameId_n=0$, in spite of the fact that the only root of the characteristic polynomial of $operatornameId_n$ has multiplicity $n$. So, no, you don't have to prove that all roots of the characteristic polynomial of $A$ have multiplicity $1$.
    $endgroup$
    – José Carlos Santos
    Apr 4 at 20:37










  • $begingroup$
    yes , you're right i edit my mistake
    $endgroup$
    – JoshuaK
    Apr 4 at 20:39










  • $begingroup$
    What does "when $A$ is written this way" mean?
    $endgroup$
    – anomaly
    Apr 5 at 0:42












  • 2




    $begingroup$
    Note that if $A=operatornameId_n$, then $A^3-3A^2-A+3operatornameId_n=0$, in spite of the fact that the only root of the characteristic polynomial of $operatornameId_n$ has multiplicity $n$. So, no, you don't have to prove that all roots of the characteristic polynomial of $A$ have multiplicity $1$.
    $endgroup$
    – José Carlos Santos
    Apr 4 at 20:37










  • $begingroup$
    yes , you're right i edit my mistake
    $endgroup$
    – JoshuaK
    Apr 4 at 20:39










  • $begingroup$
    What does "when $A$ is written this way" mean?
    $endgroup$
    – anomaly
    Apr 5 at 0:42







2




2




$begingroup$
Note that if $A=operatornameId_n$, then $A^3-3A^2-A+3operatornameId_n=0$, in spite of the fact that the only root of the characteristic polynomial of $operatornameId_n$ has multiplicity $n$. So, no, you don't have to prove that all roots of the characteristic polynomial of $A$ have multiplicity $1$.
$endgroup$
– José Carlos Santos
Apr 4 at 20:37




$begingroup$
Note that if $A=operatornameId_n$, then $A^3-3A^2-A+3operatornameId_n=0$, in spite of the fact that the only root of the characteristic polynomial of $operatornameId_n$ has multiplicity $n$. So, no, you don't have to prove that all roots of the characteristic polynomial of $A$ have multiplicity $1$.
$endgroup$
– José Carlos Santos
Apr 4 at 20:37












$begingroup$
yes , you're right i edit my mistake
$endgroup$
– JoshuaK
Apr 4 at 20:39




$begingroup$
yes , you're right i edit my mistake
$endgroup$
– JoshuaK
Apr 4 at 20:39












$begingroup$
What does "when $A$ is written this way" mean?
$endgroup$
– anomaly
Apr 5 at 0:42




$begingroup$
What does "when $A$ is written this way" mean?
$endgroup$
– anomaly
Apr 5 at 0:42










3 Answers
3






active

oldest

votes


















3












$begingroup$

The polynomial $P(X)=X^3-3X^2-X+3 = (X-1)(X-3)(X+1)$ has three distincts real roots and you have $P(A)=0$, so $A$ is diagonalizable over $mathbbR$.






share|cite|improve this answer









$endgroup$








  • 3




    $begingroup$
    I think you should provide more explanation for how you go from "three [distinct] real roots" to "diagonalizable".
    $endgroup$
    – Acccumulation
    Apr 4 at 23:09










  • $begingroup$
    @Acccumulation In my country, when we learn things about diagonalization, one of the first criterias we learn is that "if there exists a polynomial $P$ with distincts simple zeros such that $P(A)=0$, then $A$ is diagonalizable". That's why I thought it was a well-known criteria... If you want an explanation, then one can say that the minimal polynomial has to divide this one, so it has also distinct simple zeros, and it is enough to conclude... no ?
    $endgroup$
    – TheSilverDoe
    Apr 5 at 8:14


















2












$begingroup$

Solving a simpler example, $A-cI_n=0$, it's clear that $a_(i,i)=c$ because $a_(i,j) - cI_(i,j) = 0$ for all $i,j in 1,dots,n$. From here, a slightly more complicated example is $(A-cI)(A-dI)=0$ forces two conditions (for diagonal elements of $A$ and off-diagonal elements of $A$) that will lead you to a solution for general matrix polynomials.






share|cite|improve this answer









$endgroup$




















    1












    $begingroup$

    We know that the minimal polynomial divides any polynomial that $A$ is a root of. It's pretty easy to guess that $x=1$ is a root, and using polynomial division you can find that the other two roots are $x=-1, x=3$. Since all the roots are of multiplicity $1$, all of the roots of the minimal polynomial are of multiplicity $1$, and so $A$ is diagonalizable.






    share|cite|improve this answer









    $endgroup$












    • $begingroup$
      Nice way to do it using this prorpiety , i was wondering what can i say about A being inversible
      $endgroup$
      – JoshuaK
      Apr 4 at 21:00

















    3 Answers
    3






    active

    oldest

    votes








    3 Answers
    3






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    3












    $begingroup$

    The polynomial $P(X)=X^3-3X^2-X+3 = (X-1)(X-3)(X+1)$ has three distincts real roots and you have $P(A)=0$, so $A$ is diagonalizable over $mathbbR$.






    share|cite|improve this answer









    $endgroup$








    • 3




      $begingroup$
      I think you should provide more explanation for how you go from "three [distinct] real roots" to "diagonalizable".
      $endgroup$
      – Acccumulation
      Apr 4 at 23:09










    • $begingroup$
      @Acccumulation In my country, when we learn things about diagonalization, one of the first criterias we learn is that "if there exists a polynomial $P$ with distincts simple zeros such that $P(A)=0$, then $A$ is diagonalizable". That's why I thought it was a well-known criteria... If you want an explanation, then one can say that the minimal polynomial has to divide this one, so it has also distinct simple zeros, and it is enough to conclude... no ?
      $endgroup$
      – TheSilverDoe
      Apr 5 at 8:14















    3












    $begingroup$

    The polynomial $P(X)=X^3-3X^2-X+3 = (X-1)(X-3)(X+1)$ has three distincts real roots and you have $P(A)=0$, so $A$ is diagonalizable over $mathbbR$.






    share|cite|improve this answer









    $endgroup$








    • 3




      $begingroup$
      I think you should provide more explanation for how you go from "three [distinct] real roots" to "diagonalizable".
      $endgroup$
      – Acccumulation
      Apr 4 at 23:09










    • $begingroup$
      @Acccumulation In my country, when we learn things about diagonalization, one of the first criterias we learn is that "if there exists a polynomial $P$ with distincts simple zeros such that $P(A)=0$, then $A$ is diagonalizable". That's why I thought it was a well-known criteria... If you want an explanation, then one can say that the minimal polynomial has to divide this one, so it has also distinct simple zeros, and it is enough to conclude... no ?
      $endgroup$
      – TheSilverDoe
      Apr 5 at 8:14













    3












    3








    3





    $begingroup$

    The polynomial $P(X)=X^3-3X^2-X+3 = (X-1)(X-3)(X+1)$ has three distincts real roots and you have $P(A)=0$, so $A$ is diagonalizable over $mathbbR$.






    share|cite|improve this answer









    $endgroup$



    The polynomial $P(X)=X^3-3X^2-X+3 = (X-1)(X-3)(X+1)$ has three distincts real roots and you have $P(A)=0$, so $A$ is diagonalizable over $mathbbR$.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Apr 4 at 20:39









    TheSilverDoeTheSilverDoe

    5,455216




    5,455216







    • 3




      $begingroup$
      I think you should provide more explanation for how you go from "three [distinct] real roots" to "diagonalizable".
      $endgroup$
      – Acccumulation
      Apr 4 at 23:09










    • $begingroup$
      @Acccumulation In my country, when we learn things about diagonalization, one of the first criterias we learn is that "if there exists a polynomial $P$ with distincts simple zeros such that $P(A)=0$, then $A$ is diagonalizable". That's why I thought it was a well-known criteria... If you want an explanation, then one can say that the minimal polynomial has to divide this one, so it has also distinct simple zeros, and it is enough to conclude... no ?
      $endgroup$
      – TheSilverDoe
      Apr 5 at 8:14












    • 3




      $begingroup$
      I think you should provide more explanation for how you go from "three [distinct] real roots" to "diagonalizable".
      $endgroup$
      – Acccumulation
      Apr 4 at 23:09










    • $begingroup$
      @Acccumulation In my country, when we learn things about diagonalization, one of the first criterias we learn is that "if there exists a polynomial $P$ with distincts simple zeros such that $P(A)=0$, then $A$ is diagonalizable". That's why I thought it was a well-known criteria... If you want an explanation, then one can say that the minimal polynomial has to divide this one, so it has also distinct simple zeros, and it is enough to conclude... no ?
      $endgroup$
      – TheSilverDoe
      Apr 5 at 8:14







    3




    3




    $begingroup$
    I think you should provide more explanation for how you go from "three [distinct] real roots" to "diagonalizable".
    $endgroup$
    – Acccumulation
    Apr 4 at 23:09




    $begingroup$
    I think you should provide more explanation for how you go from "three [distinct] real roots" to "diagonalizable".
    $endgroup$
    – Acccumulation
    Apr 4 at 23:09












    $begingroup$
    @Acccumulation In my country, when we learn things about diagonalization, one of the first criterias we learn is that "if there exists a polynomial $P$ with distincts simple zeros such that $P(A)=0$, then $A$ is diagonalizable". That's why I thought it was a well-known criteria... If you want an explanation, then one can say that the minimal polynomial has to divide this one, so it has also distinct simple zeros, and it is enough to conclude... no ?
    $endgroup$
    – TheSilverDoe
    Apr 5 at 8:14




    $begingroup$
    @Acccumulation In my country, when we learn things about diagonalization, one of the first criterias we learn is that "if there exists a polynomial $P$ with distincts simple zeros such that $P(A)=0$, then $A$ is diagonalizable". That's why I thought it was a well-known criteria... If you want an explanation, then one can say that the minimal polynomial has to divide this one, so it has also distinct simple zeros, and it is enough to conclude... no ?
    $endgroup$
    – TheSilverDoe
    Apr 5 at 8:14











    2












    $begingroup$

    Solving a simpler example, $A-cI_n=0$, it's clear that $a_(i,i)=c$ because $a_(i,j) - cI_(i,j) = 0$ for all $i,j in 1,dots,n$. From here, a slightly more complicated example is $(A-cI)(A-dI)=0$ forces two conditions (for diagonal elements of $A$ and off-diagonal elements of $A$) that will lead you to a solution for general matrix polynomials.






    share|cite|improve this answer









    $endgroup$

















      2












      $begingroup$

      Solving a simpler example, $A-cI_n=0$, it's clear that $a_(i,i)=c$ because $a_(i,j) - cI_(i,j) = 0$ for all $i,j in 1,dots,n$. From here, a slightly more complicated example is $(A-cI)(A-dI)=0$ forces two conditions (for diagonal elements of $A$ and off-diagonal elements of $A$) that will lead you to a solution for general matrix polynomials.






      share|cite|improve this answer









      $endgroup$















        2












        2








        2





        $begingroup$

        Solving a simpler example, $A-cI_n=0$, it's clear that $a_(i,i)=c$ because $a_(i,j) - cI_(i,j) = 0$ for all $i,j in 1,dots,n$. From here, a slightly more complicated example is $(A-cI)(A-dI)=0$ forces two conditions (for diagonal elements of $A$ and off-diagonal elements of $A$) that will lead you to a solution for general matrix polynomials.






        share|cite|improve this answer









        $endgroup$



        Solving a simpler example, $A-cI_n=0$, it's clear that $a_(i,i)=c$ because $a_(i,j) - cI_(i,j) = 0$ for all $i,j in 1,dots,n$. From here, a slightly more complicated example is $(A-cI)(A-dI)=0$ forces two conditions (for diagonal elements of $A$ and off-diagonal elements of $A$) that will lead you to a solution for general matrix polynomials.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Apr 4 at 20:44









        EricEric

        613




        613





















            1












            $begingroup$

            We know that the minimal polynomial divides any polynomial that $A$ is a root of. It's pretty easy to guess that $x=1$ is a root, and using polynomial division you can find that the other two roots are $x=-1, x=3$. Since all the roots are of multiplicity $1$, all of the roots of the minimal polynomial are of multiplicity $1$, and so $A$ is diagonalizable.






            share|cite|improve this answer









            $endgroup$












            • $begingroup$
              Nice way to do it using this prorpiety , i was wondering what can i say about A being inversible
              $endgroup$
              – JoshuaK
              Apr 4 at 21:00















            1












            $begingroup$

            We know that the minimal polynomial divides any polynomial that $A$ is a root of. It's pretty easy to guess that $x=1$ is a root, and using polynomial division you can find that the other two roots are $x=-1, x=3$. Since all the roots are of multiplicity $1$, all of the roots of the minimal polynomial are of multiplicity $1$, and so $A$ is diagonalizable.






            share|cite|improve this answer









            $endgroup$












            • $begingroup$
              Nice way to do it using this prorpiety , i was wondering what can i say about A being inversible
              $endgroup$
              – JoshuaK
              Apr 4 at 21:00













            1












            1








            1





            $begingroup$

            We know that the minimal polynomial divides any polynomial that $A$ is a root of. It's pretty easy to guess that $x=1$ is a root, and using polynomial division you can find that the other two roots are $x=-1, x=3$. Since all the roots are of multiplicity $1$, all of the roots of the minimal polynomial are of multiplicity $1$, and so $A$ is diagonalizable.






            share|cite|improve this answer









            $endgroup$



            We know that the minimal polynomial divides any polynomial that $A$ is a root of. It's pretty easy to guess that $x=1$ is a root, and using polynomial division you can find that the other two roots are $x=-1, x=3$. Since all the roots are of multiplicity $1$, all of the roots of the minimal polynomial are of multiplicity $1$, and so $A$ is diagonalizable.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Apr 4 at 20:41









            GSoferGSofer

            8831314




            8831314











            • $begingroup$
              Nice way to do it using this prorpiety , i was wondering what can i say about A being inversible
              $endgroup$
              – JoshuaK
              Apr 4 at 21:00
















            • $begingroup$
              Nice way to do it using this prorpiety , i was wondering what can i say about A being inversible
              $endgroup$
              – JoshuaK
              Apr 4 at 21:00















            $begingroup$
            Nice way to do it using this prorpiety , i was wondering what can i say about A being inversible
            $endgroup$
            – JoshuaK
            Apr 4 at 21:00




            $begingroup$
            Nice way to do it using this prorpiety , i was wondering what can i say about A being inversible
            $endgroup$
            – JoshuaK
            Apr 4 at 21:00



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            Cvu4k3,S1ZrQKmMeQSg3R8cCb4sGGOc5Q6JW4 K3C6KkwqMybETHczuwWVsW,I VgM8P5FTV,n ysjl

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