A question on Gershgorin disk Suppose that the $n$ Gershgorin discs of $A \in {\mathbb{C}^{n \times n}}$ are 
mutually disjoint.
If $A $ has real main diagonal entries and
its characteristic polynomial has only real coefficients 
Why is every eigenvalue of $A$ real?

Hint: if your polynomial has only real coefficients, then if $a+bi$ is a root, then so is $a-bi$.  If $a+bi$ is in a disk centered at a real number, is $a-bi$ also in that disk?  Can you have two values in the same disk when the disks are disjoint?
