Gerschgorin's theorem , one of the most serviceable estimates for the eigenvalues of a matrix, asserts that the largest row sum of absolute values of the matrix elements forms an upper bound for the absolute values of the eigenvalues. It results from applying the triangle inequality to the definition,
once it has been written out in terms of the matrix elements,
Since not all the components of an eigenvector are zero and at least one of them is larger than (or at most equal to) the others, any inequality in which the largest component appears in the denominator can be further improved by increasing all the ratios to 1, leaving
Inasmuch as the identity of the largest component is usually unknown, the worst case has to suffice, giving the result which is usually quoted. Sometimes the diagonal element is associated with the eigenvalue, leading to a collection of disks centered on the diagonal elements whose radii are gotten from summing the off diagonal elements in the row.
Harold V. McIntosh