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Eigenvalues and Eigenvectors of 3x3 matricies

  1. Jun 5, 2009 #1
    Hello

    Im trying to find the eigenvalues and eigenvectors of 3x3 matricies, but when i take the determinant of the char. eqn (A - mI), I get a really horrible polynomial and i dont know how to minipulate it to find my three eigenvalues.

    Can someone please help..
    Thanks
     
  2. jcsd
  3. Jun 6, 2009 #2

    CompuChip

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    There is usually nothing fancy about it, you just need some way to solve the cubic equation. Often it is possible to guess one solution, divide out a factor and solve the resulting quadratic equation. Otherwise there may be other tricks available, or you might need some general formula (the degree 3 equivalent of the quadratic "abc" formula).

    Perhaps it helps if you post an example of a matrix and your result for the characteristic equation where you have doubts?
     
  4. Jun 6, 2009 #3

    HallsofIvy

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    Cubic equations, in general, are difficult.

    If the equation has integer coefficients, you can try to use the 'rational root theorem': if m/n is a a rational root of the polynomial equation, [itex]ax^n+ \cdot\cdot\cdot+ bx+ c= 0[/itex], with integer coefficients, then the denominator, n, must divide the leading coefficient, a, and the numerator, m, must divide the constant term, c. You can factor those two cofficients to find all possible rational roots, then put them into the equation to see if they work.

    Of course, it is possible that a cubic equation does not have any rational roots. In that case, you would have to use "Cardano's cubic formula"

    I don't see how we can say more without seeing the specific matrix or cubic equation you are talking about.
     
    Last edited: Jun 7, 2009
  5. Jun 15, 2009 #4
    Try diagonalizing the matrix... Look for similarity transformation... But it is not easier than characteristic polynomial anyway. But at least you stay with matrices instead of polynomialshence, a little bit more fun.
     
  6. Jun 16, 2009 #5
    Try factoring the determinant into linear factors?
     
  7. Jun 16, 2009 #6

    Pengwuino

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    There's a couple of neat properties when finding eigenvalues that might help. For one, the trace of your matrix is equal to the sum of your eigenvalues. The determinant is equal to the product of the eigenvalues as well.
     
  8. Jun 16, 2009 #7

    HallsofIvy

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    So give us some examples of polynomial equations you are having troulbe with.
     
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