Symmetric matrix real eigenvalues

  • Thread starter phrygian
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  • #1
phrygian
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Homework Statement



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


Homework Equations



The problem says to use the quadratic formula.

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
 

Answers and Replies

  • #2
clamtrox
938
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You can write the square root term as a sum of squares, which is always positive.
 
  • #3
phrygian
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How do you deal with the -4ad term? I tried to factor it but couldn't figure out how
 
  • #4
clamtrox
938
9
Can you see what (a+d)^2 - 4ad is?
 

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