In gerschgorin's theorem, the diagonal entries $A_{ii}$ of the matrix are the eigenvalue estimates, and the radii $r_i$ of the Gerschgorin disks are corresponding error bounds. Thus $\min_i A_{ii}-r_i$ is a lower bound on the eigenvalues, and $\max_i A_{ii}+r_i$ is an upper bound.

Note that these bounds are generally poor unless the off-diagonal entries are tiny. in the latter case, one can get excellent bounds by an appropriate prior similarity transform of the matrix with as suitable scaling matrix.

To get good bounds for a general matrix, one must compute an approximate eigensystem and then express the matrix in this basis by a simiarity transform. This doesn't change the eigenvalues but makes the off-diagonal entries small, so that the above applies.

In Gerschgorin's theorem, the diagonal entries $A_{ii}$ of the matrix are the eigenvalue estimates, and the radii $r_i$ of the Gerschgorin disks are corresponding error bounds. Thus $\min_i A_{ii}-r_i$ is a lower bound on the eigenvalues, and $\max_i A_{ii}+r_i$ is an upper bound.

Note that these bounds are generally poor unless the off-diagonal entries are small. In the latter case, one can get excellent bounds by an appropriate prior similarity transform of the matrix with as suitable scaling matrix.

To get good bounds for a general matrix, one must compute an approximate eigensystem and then express the matrix in this basis by a simiarity transform. This doesn't change the eigenvalues but makes the off-diagonal entries small, so that the above applies.