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Math Help - Help on a matrix problem.

  1. #1
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    Help on a matrix problem.

    'The Matrix


    A = ( -1 1 -1)
    ( 0 c 0)
    (3+2c-c^2 b 3)

    can only be diagonalised for some values of the parameters b and c . Construct a table
    showing the values of b and c for which diagonalisation is possible. You must show
    your working.'


    For this I have done the following:

    eigenvectors -1

    P(ƛ) = -(c -ƛ)((-1-ƛ)(3-ƛ)+3+2c-c^2)

    For what I finally got

    c-ƛ =0 -> ƛ =c

    Then ƛ^2 -2ƛ +2C -ƛ^2=0

    and then getting the value for ƛ

    ƛ = 1+-(1-c)


    -----

    Is this right?
    And what should I do next with this problem?

    Thank you for your time.
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  2. #2
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    Your work so far is correct. At least, those are the eigenvalues I get. You'll notice there are only two eigenvalues. What are the algebraic multiplicities of those two eigenvalues?
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  3. #3
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    Quote Originally Posted by Ackbeet View Post
    Your work so far is correct. At least, those are the eigenvalues I get. You'll notice there are only two eigenvalues. What are the algebraic multiplicities of those two eigenvalues?
    So I need to find the value 'k' ?
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  4. #4
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    If by 'k' you mean the integer k such that

    \det(A-\lambda I)=(\lambda-2+c)(\lambda-c)^{k}F,

    I would agree with you. You haven't defined k otherwise, so I don't know what it is yet.
    Last edited by Ackbeet; October 28th 2010 at 06:07 AM. Reason: Different constant.
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  5. #5
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    Quote Originally Posted by Ackbeet View Post
    If by 'k' you mean the integer k such that

    \det(A-\lambda I)=(\lambda-2+c)(\lambda-c)^{k}F,

    I would agree with you. You haven't defined k otherwise, so I don't know what it is yet.
    Yea, that's what I meant. I kinda understand this. But not sure if I know what should I do next.
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  6. #6
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    Well, for the determinant that defines the characteristic equation, I get

    -(c-\lambda)^{2}(\lambda-2+c).

    So what is k?
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  7. #7
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    Thank you.

    When the integer k is acquired is there something that still that needs to be done for the problem?
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  8. #8
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    Yes. Is there a criterion you know of whereby you can say that a matrix is diagonalizable or not?
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  9. #9
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    Quote Originally Posted by Ackbeet View Post
    Yes. Is there a criterion you know of whereby you can say that a matrix is diagonalizable or not?
    It says:

    "The Matrix can only be diagonalised for some values of the parameters b and c. Construct a table showing the values of b and c for which diagonalisation is possible."
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  10. #10
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    That's in your problem statement. But you haven't answered my question.

    Independent of this problem, what criterion do you know of that will tell you whether a matrix is diagonalizable or not?
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  11. #11
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    Quote Originally Posted by Ackbeet View Post
    That's in your problem statement. But you haven't answered my question.

    Independent of this problem, what criterion do you know of that will tell you whether a matrix is diagonalizable or not?
    Is this when there exists an invertible matrix P such that P −1AP ?
    The matrix can be written in form A=PDP^(-1)
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  12. #12
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    That is the definition of diagonalizable, I grant you. However, it's not much use in the present circumstance, I'm afraid. Have you seen this one:

    A matrix A is diagonalizable if and only if, for every eigenvalue of A, its geometric multiplicity equals its algebraic multiplicity.

    You always know that the geometric multiplicity of an eigenvalue is less than or equal to the algebraic multiplicity. Here, we're saying that they have to be equal for all the eigenvalues. If that happens, you can diagonalize the matrix.

    So, how can you use this idea to find out when your matrix is diagonalizable?
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  13. #13
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    Quote Originally Posted by Ackbeet View Post
    That is the definition of diagonalizable, I grant you. However, it's not much use in the present circumstance, I'm afraid. Have you seen this one:

    A matrix A is diagonalizable if and only if, for every eigenvalue of A, its geometric multiplicity equals its algebraic multiplicity.

    You always know that the geometric multiplicity of an eigenvalue is less than or equal to the algebraic multiplicity. Here, we're saying that they have to be equal for all the eigenvalues. If that happens, you can diagonalize the matrix.

    So, how can you use this idea to find out when your matrix is diagonalizable?
    Is that then eigenvalue c has to equal the value k?

    c= k


    Cheers
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  14. #14
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    No, no. Try to solve the equation (A-cI)x=0, and see what you get. You're trying to find eigenvectors now, and you want to know when you need two vectors, and when you need one.
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