Gershgorin discs: localizing for free
The Gershgorin circle theorem localizes every eigenvalue without solving the characteristic polynomial. For each row i, draw the disc centered at the diagonal entry a_ii with radius R_i = sum of |a_ij| over j != i (the off-diagonal absolute row sum). Every eigenvalue lies in the union of these n discs.
A = [ 5.0, 0.3, -0.2;
0.1, 8.0, 0.4;
-0.2, 0.1, 1.0 ]
Disc 1: center 5.0, radius |0.3|+|-0.2| = 0.5 -> [4.5, 5.5]
Disc 2: center 8.0, radius |0.1|+|0.4| = 0.5 -> [7.5, 8.5]
Disc 3: center 1.0, radius |-0.2|+|0.1| = 0.3 -> [0.7, 1.3]
The discs are disjoint, so each holds exactly ONE eigenvalue.
Conclusion before any factorization: lambda's are near 5, 8, 1.Weyl's theorem: symmetric perturbations are gentle
For symmetric (Hermitian) matrices the spectral theorem guarantees real eigenvalues, and they move no more than the perturbation itself. Weyl's theorem says: if A and A+E are symmetric with sorted eigenvalues, then |lambda_k(A+E) - lambda_k(A)| <= ||E||_2 for every k. The condition number of the eigenvalues is exactly 1 — perfectly conditioned.
A companion is the Cauchy interlacing theorem: deleting a row and the matching column from a symmetric A produces a smaller matrix B whose eigenvalues interlace those of A, lambda_k(A) >= lambda_k(B) >= lambda_(k+1)(A). The same interlacing structure governs singular-value perturbation: |sigma_k(A+E) - sigma_k(A)| <= ||E||_2.
Bauer-Fike: the price of non-normality
What about general (non-symmetric) matrices? The Bauer-Fike theorem gives the answer for a diagonalizable A = V D V^-1: every eigenvalue mu of A+E satisfies min over lambda(A) of |mu - lambda| <= kappa(V) * ||E||_2, where kappa(V) = ||V|| * ||V^-1|| is the condition number of the eigenvector matrix.