Weyl's inequality

Remarks

The proof uses the Courant–Fischer min-max theorem. The span of the eigenvectors for $\lambda_i(A), \dots, \lambda_n(A)$ and the span of those for $\lambda_j(B), \dots, \lambda_n(B)$ meet in a subspace of dimension at least $n - i - j + 2$. On this subspace the Rayleigh quotient of $A + B$ is at most $\lambda_i(A) + \lambda_j(B)$, and the min-max formula gives the bound.

Taking $j = 1$ and applying the result also to $(A + B) + (-B)$ gives $$\lambda_k(A) + \lambda_n(B) \le \lambda_k(A + B) \le \lambda_k(A) + \lambda_1(B).$$ Hence $\abs{\lambda_k(A + B) - \lambda_k(A)} \le \norm{B}_2$, where $\norm{B}_2$ is the operator norm induced by the Euclidean norm.

Example: $A = \operatorname{diag}(1, 0)$ and $B = \operatorname{diag}(0, 1)$ give $A + B = I_2$, and $\lambda_2(A + B) = 1 \le \lambda_1(A) + \lambda_2(B) = 1$, with equality.

Going deeper

For a student

The corollary $\abs{\lambda_k(A + B) - \lambda_k(A)} \le \norm{B}_2$ says that the eigenvalues of Hermitian matrices are stable under perturbation. For non-Hermitian matrices this fails, and bounds such as the Bauer–Fike theorem are needed instead. Weyl's inequalities are among the inequalities in Horn's problem on the eigenvalues of a sum of Hermitian matrices.