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.
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)
I'm not sure where to go from here. Any help would be greatly appreciated!