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Diganolizing a Matrix?

  1. Mar 21, 2009 #1
    1. The problem statement, all variables and given/known data

    Let Matrix A =

    [2, 1, 0, 2]
    [-1, 0, -1, 0]
    [2, 1, 0, 1]
    [1, 0, -1, 1]

    Find it's transitional matrix C and diagonal matrix D such that A = CDC^-1.

    2. Relevant equations

    3. The attempt at a solution

    I find the determinant of A-tI and set it equal to 0 to get the characteristic polynomial: t^4 - 3x^3 + 3x^2 - 2. How can I quickly factor polynomials of degree 3 or 4 like this to solve for the eigenvalues? We are not allowed to use calculators or programs on our exam, and we are limited on time. The rest of the process I understand. Thanks in advance for the help.
  2. jcsd
  3. Mar 21, 2009 #2


    Staff: Mentor

    If (t - a) is a linear factor of your polynomial, a has to be a divisor of -2. Using polynomial long division check (t - 1), (t + 1), (t - 2), and (t + 2). If one of these works, you'll be left with a third degree polynomial, which you can run the same process on. If you can get another linear factor, you'll have two linear factors and a quadratic, which you can factor by a number of techniques.
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