Suppose V is a vector space over the scalar field F, and T is a linear operator on V. Can T have an eigenvalue l such that l is not an element of F?
I am working through Linear Algebra Done Right, 3 ed, by Sheldon Axler. Definition 5.5 on page 134 defines eigenvalues of T as elements of the scalar field F that V exists over. The definition seems to suggest that eigenvalues must always be elements of this field.
When checking my work on a problem in this section, I read the following question:
In the above question, the accepted answer claims that a linear operator on a vector space over the real numbers can have a complex eigenvalue.
Is this correct? Is the idea of a complex eigenvalue well defined for such a linear operator?