I have two eigenvalues that are given as functions of two parameters $a$ and $b$
$$\lambda_1 = - a -\sqrt{ a ^2-b^2}, \lambda_2 =- a + \sqrt{ a^2-b^2}$$
I have a restriction that $a \gt b > 0$.
I have been trying all sorts of things to determine places where the signs are the same (both negative, both positive, different signs, etc.), but nothing has worked.
For example, I tried numerical approaches, but that is kludgy
Table[{a,b,-a-Sqrt[a^2-b^2],-a+Sqrt[a^2-b^2]},{a,2,4},{b,0.1,2}]//N
I tried to compare contour plots, but I cannot seem to make use of them
ContourPlot[-a - Sqrt[a^2 - b^2], {a, 0, 5}, {b, 0, 5}]
I also tried a simplify and reduce approach using
FullSimplify[{a > b > 0, -a + Sqrt[a^2 - b^2] > 0}, a > 0 && b > 0]
FullSimplify[{a > b > 0, -a - Sqrt[a^2 - b^2] > 0}, a > 0 && b > 0]
Reduce[{a > b > 0, -a + Sqrt[a^2 - b^2] < 0}, {a, b}]
Results are not satisfying and not producing a useful result.
I tried using Manipulate with StreamPlot, which shows what I want to see, but is still kludgy
Manipulate[StreamPlot[{y,-b^2 x - 2 a y},{x,-5,5},{y,-5,5},PlotLabel->Row[{"a = ",a," , b = ",b}]],{a,0,5},{b,0,5}]
Is there some elegant way to display a comparison between those two expressions that shows the sign relationship of them and particularly if there are places where they switch sign?
Maybe I am missing something basic!