Eigenvalues of matrix not giving imaginary parts

Question

This might be a very simple problem, but I can't seem to figure out why I am getting this. I am trying to find the eigenvalues of the matrix:

{{3.9999999999998025*^14 + 0.001*I, 3.141592653589793 -3.1405926535897932*I}, 
 {3.141592653589793 - 3.1405926535897932*I, 3.9999999999998025*^14 + 0.001*I}}

but when I call the Eigenvalues function on it, I only get the real parts.

I did it on Wolfram Alpha and got the imaginary parts and real parts. Also, if I change the *10^14 to *10^13, the eigenvalues have an imaginary part.

I'm guessing it's some kind of precision problem, but I'm not sure how to resolve this. Any ideas?

Answer 1

You are correct to say that this is a problem with the precision of the numbers involved. You can set the precision of those numbers explicitly:

SetPrecision[{{3.9999999999998025*^14 + 0.001*I, 
    3.141592653589793 - 3.1405926535897932*I},
   {3.141592653589793 - 3.1405926535897932*I, 
    3.9999999999998025*^14 + 0.001*I}},
  20];

Eigenvalues[%]

(* Out: {3.9999999999998339159*10^14 - 3.1395926535897932261 I, 
 3.9999999999997710841*10^14 + 3.1415926535897932262 I} *)

It may not be relevant to your application, but you might want to consider specifying those values as infinite-precision values. For instance, instead of writing 3.141592... you could use Pi, instead of 0.001 you could use 1/1000, etc.