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Math Help - eigenvalues

  1. #1
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    eigenvalues

    Let A be a 2x2 matrix and let

    char poly(A) = z^2 - Tz + D = 0.

    T=trace(A) D=det(A)

    Show that all eigenvalues of A have nonzero real part if T*D /= 0

    Show that all eigenvalues of A have real part < 0 if D > 0 and T < 0

    Based on the characteristic polynomial of A, I know that
    T= a1 + a2, where a1 and a2 are eigenvalues of A
    D=a1*a2

    However, I do not know how to proceed from here. Any help would be greatly appreciated!!
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  2. #2
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    You use the quadratic formula to show that all eigenvalues of a 2 \times 2 have the form
    z = \frac{\text{tr}(A) \pm \sqrt{\text{tr}(A) - 4\det(A)}}{2}.
    This will show you that if the trace is zero then you have zero real part if \det(A) \ge 0. However, if you look at z^2 - \text{tr}(A) z = 0 for \det(A) = 0 you should notice that this implies that z = \{0, \text{tr(A)}\}.
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  3. #3
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    Thank you! I need to do the same with the 3x3 matrix case, but the characteristic polynomial is a cubic instead of a quadratic.

    Let A be a 3x3 matrix.

    char poly(A) = z^3 -Tz^2 + Mz - D = 0

    Show that all eigenvalues of A have nonzero real part if (TM - D)D /= 0

    Show that all eigenvalues of A are < 0 if D < 0, T < 0, and TM < D

    T = tr(A) , D = det(A), and there's not a given definition for M.

    Again, if a1, a2, a3 are eigenvalues of A, I found that
    T = a1 +a2 + a3
    M = a1*a2 + a2*a3 + a1*a3
    D = a1*a2*a3
    TM - D = (a1 + a2)*(a1 + a3)*(a2 + a3)
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