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Symmetric matrix real eigenvalues

  1. Dec 2, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove a symmetric (2x2) matrix always has real eigenvalues. The problem shows the matrix as {(a,b),(b,d)}.

    2. Relevant equations

    The problem says to use the quadratic formula.

    3. The attempt at a solution

    From the determinant I get (a-l)(d-l) - b^2 = 0 which expands to l^2 - (a+d)l + (ad - b^2) = 0

    Using the quadratic formula I get for under the square root: (a + d)^2 - 4(ad-b^2)
    How can I show that this is always positive?

    Thanks for the help
  2. jcsd
  3. Dec 2, 2009 #2
    You can write the square root term as a sum of squares, which is always positive.
  4. Dec 2, 2009 #3
    How do you deal with the -4ad term? I tried to factor it but couldn't figure out how
  5. Dec 2, 2009 #4
    Can you see what (a+d)^2 - 4ad is?
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