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Finding the eigenvalues of a 3x3 matrix

  1. Mar 30, 2014 #1
    1. The problem statement, all variables and given/known data
    A =
    7 -5 0
    -5 7 0
    0 0 -6

    Can you please show your method aswell. Every time I try I get the wrong answer.

    FYI Eigen values are 12.2,-6


    3. The attempt at a solution
    so far I got:
    det =
    7-λ -5 0
    -5 7-λ 0
    0 0 -6-λ

    Im unsure what to do next. I tried doing this
    (7-λ) [(7-λ)*(-6-λ)] - (-5)(-5(-6-λ)) + 0 = 0
    but when I expand and get a cubic equation and solve it, i dont get the right answer which is 12,2,-6
     
  2. jcsd
  3. Mar 30, 2014 #2

    SteamKing

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    Show us your cubic equation in λ after you have done the algebra and collected all the terms. You may have made a mistake in your calculations.
     
  4. Mar 30, 2014 #3
    continuing from (7-λ) [(7-λ)*(-6-λ)] - (-5)(-5(-6-λ)) + 0 = 0


    (7-λ) [(7-λ)*(-6-λ)] + 5 (30+5λ) = 0
    (7-λ) [(7-λ)*(-6-λ)] +150 + 25λ = 0
    (7-λ) [-42-λ+λ^2] + 150 + 25λ = 0

    294 - 7λ + 7λ^2 + 42λ + λ^2 - λ^3 + 150 +25λ = 0

    444 + 60λ +8λ^2 - λ^3 = 0

    So i checked that with the calculator and the answer is completely off.
    Also in my first semester exam we aren't allowed to use the calc. so i need to find another way to solve the cubic equation OR instead do some sort of manipulation of the initial det matrix and get the answer from there.

    Any ideas?
    and thank you
     
  5. Mar 30, 2014 #4
    Looks to me like you should have:

    -144 + 60λ +8λ^2 - λ^3 = 0
     
  6. Mar 30, 2014 #5

    SteamKing

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    Check your cubic polynomial again. You have an arithmetic error in calculating the constant term.

    Here's a tip: when you have something like (7-λ)(7-λ)(-6-λ), evaluate the two identical terms first (7-λ)(7-λ) = 49 - 14λ + λ[itex]^{2}[/itex], since you should be able to write the square of a monomial by doing a mental calculation.

    Alternately, when you want to calculate the determinant of a matrix with several zeroes in a single row or column, like the matrix in this problem, using minors might save you considerable calculation.

    In any event, when you obtain the proper cubic characteristic equation, you can use several different methods to find the solutions: you can guess a solution, you can use the rational root theorem, or you can plot the equation.
     
  7. Mar 30, 2014 #6

    AlephZero

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    The factor (-6-λ) is in both terms. So you can simplify this to
    [ (7-λ)(7-λ) - (-5)(-5) ] (-6-λ) = 0
    and save a lot of work.

    And (7-λ)(7-λ) - (-5)(-5) is of the form ##a^2 - b^2##, so you should be able to factorize it without multiplying everything out.
     
  8. Mar 31, 2014 #7
    Oh yeah. But sill I used the values you got and I checked with an online cubic calculator, and this is what I got

    attachment.php?attachmentid=68174&stc=1&d=1396247658.png

    The answer is quite close. x1 should be 12 and x2 should be -6.

    can you explain why this is so??
     

    Attached Files:

  9. Mar 31, 2014 #8
    That is awesome. But i dont get how you were able to factorize it WITHOUT multiplying it out?
     
  10. Mar 31, 2014 #9

    Mark44

    Staff: Mentor

    AlephZero explained where the factorization comes from. If a = 7 - λ and b = -5, then (7-λ)(7-λ) - (-5)(-5) = a2 - b2, so you can factor it as (a - b)(a + b) or as (7 - λ - (-5))(7 - λ + (-5)).
     
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