Gershgorin theorem and the field of values

Question

Just a naive question, to get a better feeling for the subject.

Let A be a complex nonsingular n x n matrix.

Be W(A) the field of values of A and D(A) the union of all Gershgorin discs of A, with $R_i = min(\ \sum_{i=0}^n|a_{ij}|\ ,\ \sum_{i=0}^n|a_{ji}|\ )$.

Is $D(A) \subset W(A)$ or could be $D(A)\backslash W(A)\neq\varnothing$?

Edit: Had to change D(A) a bit.

Answer 1

If $A$ is a non-diagonal Hermitian matrix the numerical range $W(A)$ is contained in $\mathbb R$, while $D(A)$ is not.