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Homework Statement
Suppose the the matrix A is symmetric, meaning that
A =
a b
b d
Show that for any symmetric matrix A there are always real eigenvalues. Also, show that
the eigenvectors corresponding to two dierent eigenvalues are always orthogonal; that is,
if V1 and V2 are the eigenvectors for eigenvalues [tex]\lambda[/tex]1 and [tex]\lambda[/tex]2, with [tex]\lambda[/tex]1 not equal to [tex]\lambda[/tex]2, then the dot
product V1 * V2 = 0.
HINT: Compare [tex]\lambda[/tex]1V1*V2 to V1*[tex]\lambda[/tex]2V2 .
Homework Equations
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The Attempt at a Solution
I was able to show that any eigenvalue from this matrix would always be positive. That wasn't too bad. Its the second part that is tripping me up. I don't get how the hint is really all that helpful. I want to find values of V1 and V2 don't I? I'm confused.