eigenvalue estimates
Computing eigenvalues exactly means solving a polynomial, which for large matrices is impractical and for degree five or more has no formula at all. Eigenvalue estimation sidesteps this: it pins down where the eigenvalues must lie — bounds and regions — directly from the entries, without ever solving the characteristic equation.
The workhorse is Gershgorin's circle theorem. For each row i, draw a disc in the complex plane centered at the diagonal entry a_ii with radius equal to the sum of the absolute values of the other entries in that row, R_i = sum_{j != i} |a_ij|. Every eigenvalue lies in the union of these Gershgorin discs. A quick glance at a matrix thus traps its whole spectrum in a handful of circles.
For symmetric (or Hermitian) matrices, the Rayleigh quotient gives sharp variational bounds. The quotient r(x) = <Ax, x> / <x, x> always lies between the smallest and largest eigenvalues, so any test vector x produces a guaranteed lower bound on lambda_max and upper bound on lambda_min. The Courant-Fischer min-max theorem extends this to every eigenvalue in between.
Each row gives a disc centered at its diagonal entry; the union of discs traps every eigenvalue.
Gershgorin also gives a separation result: if k discs form a connected component disjoint from the rest, that component contains exactly k eigenvalues (with multiplicity). Applying the theorem to A^T as well, since A and A^T share eigenvalues, tightens the bounds using column sums.