Some glitch in the Plot: Two approaches for plotting give different results

Question

I have a matrix whose eigenvalues have to be plotted. I am using two techniques but they are not the same. Please guide me on this.

Here is first command or code used for plotting the eigenvalues(two in my case)

a := 1;
SM[j1_, j2_, k_] := {{0, -(j1 + j2*Exp[I*k*a])}, {-(j1 + (j2*Exp[-I*k*a])), 
    0}};
eigenen[j1_, j2_, k_] := Module[{EV1}, EV1 = Eigenvalues[SM[j1, j2, k]];
  Plot[EV1, {k, -\[Pi]/a, \[Pi]/a}, PlotLegends -> "Expressions"]]
Manipulate[eigenen[j1, j2, k], {j1, 0.1, 1}, {j2, 0.1, 1}]

Result is (j1>j2, you can take j1=0.3, j2=0.1)

Second used code

a:=1;
PowerExpand[FullSimplify[Eigenvalues[ {{0, -(j1 + j2*Exp[I*ka])},{-(j1 + j2*Exp[-I*ka]), 0}} ]]]

Out:= {-Sqrt[j1^2 + j2^2 + 2 j1 j2 Cos[ka]], Sqrt[j1^2 + j2^2 + 2 j1 j2 Cos[ka]]}
Manipulate[Plot[{Sqrt[(j1^2 + j2^2 + 2 j1 j2 Cos[k*a])], -Sqrt[(j1^2 + j2^2 + 2 j1 j2 Cos[k*a])]}, {k, -\[Pi]/a, \[Pi]/a}, 
  PlotLegends -> "Expressions"], {j1, 0.1, 1}, {j2, 0.1, 1}]

For this result is (j1>j2, you can take j1=0.3, j2=0.1)

I have no idea about the difference but I will be using first code more often as I later have to deal with large matrices(40$\times$ 40), where I can't simply write the eigenvalues. Is there any effective or similar code where this problem can be resolved?

My queries(to summarise):
(i) Reason(s) for the difference in the plots.
(ii) Anyway the first code can be resolved to give similar results as Ist one(those are expected ones), not as the Ist ones.

Answer 1

Here is your problem in more evident way:

a = 1;
SM[j1_, j2_, k_] = {{0, -(j1 + j2*Exp[I*k *a])}, {-(j1 + j2*Exp[-I*k* a]), 0}};

Eigenvalues[SM[0.1, 0.1, -Pi/a + 0.1]]
(*{-0.00999583, 0.00999583}*)

Eigenvalues[SM[j1, j2, k]] /. {j1 -> 0.1, j2 -> 0.1, k -> -Pi/a + 0.1}
(*{0.00999583 - 1.99493*10^-16 I, -0.00999583 + 1.99493*10^-16 I}*)

If you ignore the imaginary part from the second expression, the values are in opposite order. You either evaluate eigenvalue algorithm on numeric matrix at each point or evaluate the generic symbolic version once and make substitution. Eigenvalues can make some inner decisions in what order to return the values, so the second approach should be more general and therefore correct.