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Math Help - Finding eigenvalues of a 3x3 matrix

  1. #1
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    Finding eigenvalues of a 3x3 matrix

    I am having problems trying to orthogonally diagonalise the below matrix. I am stuck at the stage where you find the eigenvalues.


    The matrix is

    5 -4 0
    -4 3 -4 = 0
    0 -4 1

    This is what I get, but I cannot solve it so I think I have done something wrong.


    5-λ -4 0
    -4 3-
    λ -4 = 0
    0 -4 1-
    λ



    (5-
    λ)*|(3-λ)(1-λ) - (-4)(-4)| + 4*|(-4)(1-λ)| = 0


    this works down to

    (5-
    λ)(λ^2 +12λ - 29)


    when typing this I have just realised that I just need to use the quadratic equation to solve
    (λ^2 +12λ - 29) and (5 - λ) is just λ = 5.


    So if someone could just confirm if I'm heading towards the right direction or not, then that would be great


    =


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  2. #2
    Super Member ILikeSerena's Avatar
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    Re: Finding eigenvalues of a 3x3 matrix

    Hi marky82!

    I think you made a calculation error.

    In (5-λ)*|(3-λ)(1-λ) - (-4)(-4)| + 4*|(-4)(1-λ)| = 0, you dropped a factor (5-λ) in the 2nd term.

    Furthermore (5-λ) is not a factor of the entire expression.

    You are heading in the right direction though.
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    Re: Finding eigenvalues of a 3x3 matrix

    thanks for replying. I am not sure what you mean. Could you show me how you worked out the eigenvalues so I can compare it with mine please?
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  4. #4
    Super Member ILikeSerena's Avatar
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    Re: Finding eigenvalues of a 3x3 matrix

    Well, you can see the eigenvalues for instance here.
    As you can see 5 is not one of them.
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    Re: Finding eigenvalues of a 3x3 matrix

    Quote Originally Posted by ILikeSerena View Post
    Hi marky82!

    I think you made a calculation error.

    In (5-λ)*|(3-λ)(1-λ) - (-4)(-4)| + 4*|(-4)(1-λ)| = 0, you dropped a factor (5-λ) in the 2nd term.

    Furthermore (5-λ) is not a factor of the entire expression.

    You are heading in the right direction though.
    Thanks, for the link, it helps when I have answer to work towards. My maths terminology isn't the greatest so I don't really understand the above quote. I don't get what it means to have dropped a factor (5 - λ) in the 2nd term. It's probably so simple but my brain doesn't get what you mean
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    Re: Finding eigenvalues of a 3x3 matrix

    Apologies, It has just clicked. Thanks so much for your help
    Thanks from ILikeSerena
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  7. #7
    Super Member ILikeSerena's Avatar
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    Re: Finding eigenvalues of a 3x3 matrix

    Aaah! You mean the meaning of the words I've used!?
    Are you referring to term and factor?
    Or something else?
    I'd like to know for future reference.
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    Re: Finding eigenvalues of a 3x3 matrix

    Quote Originally Posted by ILikeSerena View Post
    Aaah! You mean the meaning of the words I've used!?
    Are you referring to term and factor?
    Or something else?
    I'd like to know for future reference.
    yeah, term and factor. I normally find I can do most maths processes but the terminology is what I have to spend most time on getting to grips with.
    Thanks from ILikeSerena
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  9. #9
    Super Member ILikeSerena's Avatar
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    Re: Finding eigenvalues of a 3x3 matrix

    Quote Originally Posted by marky82 View Post
    yeah, term and factor. I normally find I can do most maths processes but the terminology is what I have to spend most time on getting to grips with.
    Yeah, I know what you mean.
    Whenever I learn something new, the first challenge is always to come to grips with the new words that are being used, or otherwise with the words that I simply did not know (well enough) yet.
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  10. #10
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    Re: Finding eigenvalues of a 3x3 matrix

    Yeah its like its starting to sound like another language when I speak it out loud.
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