Visualizing Signs of Two Expressions

Question

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!

Answer 1

• RegionPlot. When a>b>0,-a + Sqrt[a^2 - b^2] and -a - Sqrt[a^2 - b^2] are both negative.

Clear[f, g];
f[a_, b_] = -a - Sqrt[a^2 - b^2];
g[a_, b_] = -a + Sqrt[a^2 - b^2];
GraphicsRow@{RegionPlot[
   f[a, b] < 0 && g[a, b] < 0, {a, 0, 4}, {b, 0, 4}, PlotPoints -> 50],
  RegionPlot[f[a, b] > 0 && g[a, b] > 0, {a, 0, 4}, {b, 0, 4}, 
   PlotPoints -> 50],
  RegionPlot[
   f[a, b] > 0 && g[a, b] < 0 || f[a, b] < 0 && g[a, b] > 0, {a, 0, 
    4}, {b, 0, 4}, PlotPoints -> 50]}

Answer 2

If I understand you correctly, try this:

    Plot3D[{-a - Sqrt[a^2 - b^2], -a + Sqrt[a^2 - b^2]}, {a, 0, 5}, {b, 0,
   5}, PlotStyle -> {{Blue, Opacity[0.5]}, {Orange, Opacity[0.5]}}, 
 AxesLabel -> {"a", "b", "expr"}]

yielding the following:

One can see that the orange surface at 0<a<5 && 0<b<5 everywhere lies over the blue one.

On the other hand, one can use Reduce as follows:

Reduce[(-a - Sqrt[a^2 - b^2] < -a + Sqrt[a^2 - b^2]) && b > 0 && a > b]

(*   b > 0 && a > b  *)

which means that provided b > 0 && a > b the inequality -a - Sqrt[a^2 - b^2] < -a + Sqrt[a^2 - b^2] always holds.

Have fun!