  Let ({\lambda}, v) be a known real eigenpair of a square real matrix A. In
this paper it is shown how to locate the other eigenvalues of A in terms of the
components of v. The obtained region is a union of Gershgorin discs of the
second type recently introduced by the authors in a previous paper. Two cases
are
  considered depending on whether or not some of the components of v are equal
to zero. Upper bounds are obtained, in two different ways, for the largest
eigenvalue in absolute value of A other than {\lambda}. Detailed examples are
provided. Although nonnegative irreducible matrices are somewhat emphasized,
the main results in this paper are valid for any square real matrix.
