I'm trying to clarify this theorem, in particular a few statements which seem contradictory.
1) If k discs are disjoint from the others, their union contains k eigenvalues. (my lecture notes)
and on the Wikipedia page it says,
2) If the union of k discs is disjoint from the union of the other n − k discs then the former union contains exactly k and the latter n − k eigenvalues of A.
But also notes
"If one of the discs is disjoint from the others then it contains exactly one eigenvalue. If however it meets another disc it is possible that it contains no eigenvalue"
In particular the 2nd statement seems to claim that if you have a union of k disjoint discs, the remaining union n - k discs have n-k eigenvalues, is that not contradicted by the fact it is possible for a union of intersecting discs with no eigenvalue as claimed above? Edit: I think I made an error in reading this, the statement actually seems to claim that the disc itself may not have one, not the union, regardless should the initial statement from my notes be changed to distinct?
If it is true that a union of intersecting discs has no guaranteed of eigenvalues, the first statement (which was given in my notes) seems to suggest if you have k disjoint discs, they have k eigenvalues. Should that statement be corrected to "k distinct eigenvalues"?
Thanks for reading.