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I'm trying to find the zeros of the eigenvalues (functions of $k_x$) of a self-adjoint $4\times4$ matrix H:

H = {{kx^2 + a kx + B, 0, I a kx, 
    del}, {0, -kx^2 + a kx - B, -del, -I a kx}, {-I a kx, -del, 
    kx^2 - a kx - B, 0}, {del, I a kx, 0, -kx^2 - a kx + B}};
B = 1;
a = 0.5;
del = 0.02;
evals = Eigenvalues[H]
solns = Table[NSolve[evals[[ii]] == 0, kx, Reals], {ii, 4}]

My eigenvalues are all real, but when mathematica calculates them, because of the complex entries in the off-diagonals, it produces a +0I inside the root on some of the coefficients. This causes a problem when I try to solve restricting the domain to the reals:

Solve::nddc: The system contains a nonreal constant 0.128451 +0. I. With the domain specified, all constants should be real.

I have tried using Re which doesn't work, and Chop which completely breaks the output when applied to the smaller numbers in the full code (I've tried multiplying by a constant and dividing afterwards). ToRadicals leads to an extremely large expression which NSolve struggles to evaluate (FullSimplify on this gives similar problems). How do I remove the +0I efficiently?

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  • $\begingroup$ Does expr /. I -> 0 work? $\endgroup$
    – Marius Ladegård Meyer
    Commented Sep 4, 2019 at 12:43
  • $\begingroup$ Provide definition of H so that we can reproduce the problem $\endgroup$
    – Bob Hanlon
    Commented Sep 4, 2019 at 13:24
  • $\begingroup$ @MariusLadegårdMeyer Nope $\endgroup$
    – Elliott Mansfield
    Commented Sep 4, 2019 at 13:28
  • $\begingroup$ @BobHanlon updated $\endgroup$
    – Elliott Mansfield
    Commented Sep 4, 2019 at 13:36
  • 1
    $\begingroup$ Please provide the definition in Mathematica code along with the values of the parameters that are causing the problem. $\endgroup$
    – Bob Hanlon
    Commented Sep 4, 2019 at 13:37

1 Answer 1

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Clear[H, B, a, del]

H = {{kx^2 + a kx + B, 0, I a kx, 
    del}, {0, -kx^2 + a kx - B, -del, -I a kx}, {-I a kx, -del, 
    kx^2 - a kx - B, 0}, {del, I a kx, 0, -kx^2 - a kx + B}};
B = 1;
a = 0.5;
del = 0.02;

Use Chop in the definition of evals to eliminate the complex artifacts (0. I).

evals = Eigenvalues[H] // Chop;

solns = NSolve[# == 0, kx, Reals] & /@ evals

(* {{}, {{kx -> -1.34174}, {kx -> -0.845824}}, {{kx -> 1.34174}, {kx -> 
    0.845824}}, {}} *)

Alternatively, use exact values for the parameters

Clear[H, B, a, del]

H = {{kx^2 + a kx + B, 0, I a kx, 
    del}, {0, -kx^2 + a kx - B, -del, -I a kx}, {-I a kx, -del, 
    kx^2 - a kx - B, 0}, {del, I a kx, 0, -kx^2 - a kx + B}};
B = 1;
a = 1/2;
del = 1/50;

evals = Eigenvalues[H];

solns2 = NSolve[# == 0, kx, Reals] & /@ evals

(* {{}, {{kx -> -1.34174}, {kx -> -0.845824}}, {{kx -> 1.34174}, {kx -> 
    0.845824}}, {}} *)

solns == solns2

(* True *)
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