Find interval/range of certain coefficients to prove conditions of eigenvalues

Question

I have seen similar questions to the one I am posting here, but I haven't been able to execute it on Mathematica. I would request some help with the following:

I have the following matrix:

{{1.79847 + 3.74119 d, -5.12392 + 4.67648 a - 
   0.935297 d, -0.199451, -0.0666941 - 
   0.935297 d, -0.341908}, {-0.213141, 1.03378, 0, 0.0236824, 0}, {0, 
  0, -0.0221797, 0, 0.010101}, {0, 1, 0, 1, 0}, {0.8 d, a - 0.2 d, 
  0, -0.2 d, 0}}

This matrix has the following eigenvalues:

{Root[0.00204575 a + 
    0.00775602 d + (-0.0163848 + 0.0487214 a + 
       0.177266 d) #1 + (-0.659423 - 1.04752 a - 
       4.14868 d) #1^2 + (3.49068 + 0.996752 a + 
       7.59996 d) #1^3 + (-3.81007 - 3.74119 d) #1^4 + #1^5 &, 1], 
 Root[0.00204575 a + 
    0.00775602 d + (-0.0163848 + 0.0487214 a + 
       0.177266 d) #1 + (-0.659423 - 1.04752 a - 
       4.14868 d) #1^2 + (3.49068 + 0.996752 a + 
       7.59996 d) #1^3 + (-3.81007 - 3.74119 d) #1^4 + #1^5 &, 2], 
 Root[0.00204575 a + 
    0.00775602 d + (-0.0163848 + 0.0487214 a + 
       0.177266 d) #1 + (-0.659423 - 1.04752 a - 
       4.14868 d) #1^2 + (3.49068 + 0.996752 a + 
       7.59996 d) #1^3 + (-3.81007 - 3.74119 d) #1^4 + #1^5 &, 3], 
 Root[0.00204575 a + 
    0.00775602 d + (-0.0163848 + 0.0487214 a + 
       0.177266 d) #1 + (-0.659423 - 1.04752 a - 
       4.14868 d) #1^2 + (3.49068 + 0.996752 a + 
       7.59996 d) #1^3 + (-3.81007 - 3.74119 d) #1^4 + #1^5 &, 4], 
 Root[0.00204575 a + 
    0.00775602 d + (-0.0163848 + 0.0487214 a + 
       0.177266 d) #1 + (-0.659423 - 1.04752 a - 
       4.14868 d) #1^2 + (3.49068 + 0.996752 a + 
       7.59996 d) #1^3 + (-3.81007 - 3.74119 d) #1^4 + #1^5 &, 5]}

I need to find conditions on coefficients 'a' and 'd' jointly such that both the 4th and 5th eigenvalues above are greater than 1 in absolute value.

If it helps in reducing your effort, the characteristic polynomial(in x) of the above matrix is the following where A and B are the coefficients:

0. - 0.00204575 a - 0.00775602 d + 0.0163848 x - 0.0487214 a x - 
 0.177266 d x + 0.659423 x^2 + 1.04752 a x^2 + 4.14868 d x^2 - 
 3.49068 x^3 - 0.996752 a x^3 - 7.59996 d x^3 + 3.81007 x^4 + 
 3.74119 d x^4 - 1. x^5

So I need to solve for 'a' and 'd', find a condition on them such that the 4th and 5th roots are bigger than 1. Thanks very much in advance.

Answer 1

With mat defined as your matrix above, you can run the commands

evals = Eigenvalues[mat];
RegionPlot[Abs[evals[[4]]] >= 1 && Abs[evals[[5]]] >= 1, {a, -5, 5}, {d, -5, 5}, 
 PlotPoints -> 100, FrameLabel -> {a, d}]

The PlotPoints -> 100 option does cause the code to take more time, but may be necessary to get higher "resolution" on the features such as that narrow slot near the origin (which does appear to be a true feature on the plot and not some weird rounding error.)

Alternately, if you want to ensure that there are at least two eigenvalues whose absolute value is greater than or equal to 1 (not necessarily the fourth and fifth ones in the above list):

RegionPlot[Sort[Abs[evals]][[-2]] >= 1, {a, -5, 5}, {d, -5, 5}, 
 PlotPoints -> 100, FrameLabel -> {a, d}]

Note that these two plots are different because Mathematica does not order the roots of a polynomial in increasing order of absolute magnitude by default. Rather, it lists all the real roots from least to greatest, and then the complex roots according to their real part (from least to greatest).