1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook