Let $M$ be the matrix representation of a complex number:
$$ M=\pmatrix{a&b\\-b&a} \implies \det M=a^2+b^2 $$
My question is what do the eigenvalues of $M$ represent?
$$ det(sI-M)=0 \implies s = \{a + ib, a- ib\} $$
For reference the eigenvectors are $v=\{ (i,1), (-i,1)\}$
The standard interpretation of the eigenvalues is that they represent the stretch of the linear transformation along the eigenbasis. But I am at a loss as to how to visualize this in the case of complex numbers. What is the linear transformation is this case: is M a transformation from the origin (0,0) to position (a,b) on the complex plane? If this is the case, why are we getting two vectors if the point (a,b) goes to a single position on the plane?