Plot artifacts depend on replacement rule

Question

I am using Plot to look at the eigenvalues of a matrix, and find that in some circumstances I get artifacts. For example

H = {{x, -g}, {g, -x}};
Eigenvalues[H]
Plot[Eigenvalues[H] /. g -> 1, {x, -4, 4}]

gives me

{-Sqrt[-g^2 + x^2], Sqrt[-g^2 + x^2]}

but

H = {{x, -1}, {1, -x}};
Eigenvalues[H]
Plot[Eigenvalues[H], {x, -4, 4}]

gives me

{-Sqrt[-1 + x^2], Sqrt[-1 + x^2]}

Fianlly, just to add to the confusion,

Plot[{-Sqrt[-1 + x^2], Sqrt[-1 + x^2]}, {x, -4, 4}]

gives

What's going on here? I thought there would be no difference between these.

Answer 1

When you are using Eigenvalues[H] inside Plot it is evaluated each time for an x, and depending on the matrix the order of the eigenvalues may change as well. Evaluate it before and then plot

H = {{x, -1}, {1, -x}};
ev = Eigenvalues[H]
Plot[ev, {x, -4, 4}]

$\left\{-\sqrt{x^2-1},\sqrt{x^2-1}\right\}$

Or you can try to Sort it

Plot[Sort[Eigenvalues[H]], {x, -4, 4}]