Why Gerschgorin Theorem just use the sum of row entries as the radius?

Question

From the Gerschgorin Theorem we know that each eigenvalue located in the circles which radius are:$$r_{i}= \sum_{j=1,j\ne i}^n | a_{ij}|$$(which means add each entries in a row except for the diagonal)$$$$But we also know that the $A^T$and A has the same eigenvalue. So we can conclude from that for each $r_{i}$,we have $$r_{j}= \sum_{i=1,i\ne j}^n | a_{ij}|$$(simply add the abs of each entries in each colum except the diagonal). So from that we can know the minimum radius of the circle of the ith eigenvalus is the minimum of $r_{i}$ and $r_{j}$. But why the textbook just told us that the minimum is $r_{i}$?