Thread: 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!!

You use the quadratic formula to show that all eigenvalues of a have the form
.
This will show you that if the trace is zero then you have zero real part if . However, if you look at for you should notice that this implies that .

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)