I still can' understand what you mean...but never mind: you wrote , which I suppose means to find the eigenvalues of a matrix
which is then . You say your book says that and all its scalar multiples are eigenvalues of A corresponding to . This is not so:
, as you can easily check.
So either you miscopied some of the given terms, or you looked at the answer to the wrong question, or your book is dead wrong.
You may now want to check that the vector I gave you in my first post really is an eigenvector of the above matrix A corresponding to ...
Trust me I understand what you are saying because I did the problem the other way original.
However, I doubled checked the answers and I didn't have a copying error and this isn't just a one time occurrence of how they handling imaginary eigenvectors and spaces.
I could understand a book having a mistake on one problem but on all complex problems would be hard to fathom as well.
Therefore, I am stuck in a quandary since it is hard to have a consistency of the same error.