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Find a 3*3 matrix A which is not diagonalizable and such that 2 is the only eigenvalu

  1. Apr 5, 2009 #1
    1. The problem statement, all variables and given/known data

    Find a 3*3 matrix A which is not diagonalizable and such that 2 is the only eigenvalue of A

    2. Relevant equations



    3. The attempt at a solution

    since λ=2,and it is a 3*3 matrix

    i get the det(λI-A)=(λ-2)^3=0

    then λ^3-6λ^2+12λ-8=0

    now i use http://image.cramster.com/answer-board/image/cramster-equation-2009452223286337456700835550002161.gifas[/URL] matirx A

    and i calculate

    det(λI-A)
    =λ^3-λ^2(i+e+a)+λ(ai+ae+ie-cg-hf-bd)-(aef+bfg+cdh+cge-ahf-bdi)=0

    now we can easy to know that

    i+e+a=6

    ai+ae+ie-cg-hf-bd=12

    aef+bfg+cdh+cge-ahf-bdi=18

    now im stuck here

    can anyone help me to slove these equtions.

    i dont konw the way i doing this problem is right or wrong.

    if u have any others ideas,plz help me
     
    Last edited by a moderator: Apr 24, 2017
  2. jcsd
  3. Apr 6, 2009 #2

    matt grime

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    Re: find a 3*3 matrix A which is not diagonalizable and such that 2 is the only eigen

    There are infinitely many solutions to those equations: 9 variables and 3 equations. Also, solutions to them tell you nothing about whether or not the matrix can be diagonalised. You're going to have to think of a different method.

    Do you know of any other forms a matrix can be put into if it is not diagonlisable?

    Do you know of any matrices where it is easy to see what the eigenvalues are?
     
  4. Apr 6, 2009 #3
    Re: find a 3*3 matrix A which is not diagonalizable and such that 2 is the only eigen

    sorry, i dont have any ideas

    could u just give me any hints?
     
  5. Apr 6, 2009 #4

    Dick

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    Re: find a 3*3 matrix A which is not diagonalizable and such that 2 is the only eigen

    Hint: think about upper triangular matrices.
     
  6. Apr 6, 2009 #5
    Re: find a 3*3 matrix A which is not diagonalizable and such that 2 is the only eigen

    but the matrix A is not diagonalizable
     
  7. Apr 6, 2009 #6

    Dick

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    Re: find a 3*3 matrix A which is not diagonalizable and such that 2 is the only eigen

    Do you really think all upper triangular matrices are diagonalizable?
     
  8. Apr 6, 2009 #7
    Re: find a 3*3 matrix A which is not diagonalizable and such that 2 is the only eigen

    ok,i trid again and stuck again

    i set A as

    a b c
    0 d e
    0 0 f

    use the way i did before above

    finally i get

    a+d+f=6
    fd+af+ad=12
    adf=18

    then.................=.=

    i still cant slove these equtions

    Dick , wut exactly do u mean by ues an upper matrix
     
  9. Apr 6, 2009 #8

    Dick

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    Re: find a 3*3 matrix A which is not diagonalizable and such that 2 is the only eigen

    By an upper triangular matrix I mean exactly what you posted. You should be able to tell me what the eigenvalues of that matrix are without working too hard. That tells you what values a, d and f should have. Now you just want to find an EXAMPLE of values of b, c and e that is NOT diagonalizable. Hint: it's not diagonalizable if you can't find three linearly independent eigenvectors. Just try guessing. Hint: b=c=e=0 is a bad guess.
     
  10. Apr 7, 2009 #9

    matt grime

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    Re: find a 3*3 matrix A which is not diagonalizable and such that 2 is the only eigen

    Where did those 3 equations come from, and why? I think you need to work out the determinant of an upper triangular matrix again. You still haven't addressed what it means for a matrix to be (or not be) diagonalizable, either.
     
  11. Apr 8, 2009 #10
    Re: find a 3*3 matrix A which is not diagonalizable and such that 2 is the only eigen

    i got it now !!!

    i really went to a wrong way!!

    thanks for DICK and MG you guys help!!!
     
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