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  "body": "\n## Statement\n\nLet $A$ and $B$ be $n \\times n$ [[Hermitian matrices|0P17NGQQMQ]].\nFor a Hermitian matrix $M$, let $\\lambda_1(M) \\ge \\dots \\ge \\lambda_n(M)$ be its [[eigenvalues|0BQAA2BK3Q]], listed with multiplicity.\nThen, for all $i, j \\ge 1$ with $i + j - 1 \\le n$,\n$$\\lambda_{i+j-1}(A + B) \\le \\lambda_i(A) + \\lambda_j(B).$$\n\n## Remarks\n\nThe proof uses the [[Courant–Fischer min-max theorem|CJK2HS2508]].\nThe 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 $n - i - j + 2$.\nOn 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.\n\nTaking $j = 1$ and applying the result also to $(A + B) + (-B)$ gives\n$$\\lambda_k(A) + \\lambda_n(B) \\le \\lambda_k(A + B) \\le \\lambda_k(A) + \\lambda_1(B).$$\nHence $\\abs{\\lambda_k(A + B) - \\lambda_k(A)} \\le \\norm{B}_2$, where $\\norm{B}_2$ is the [[operator norm|06EKW1AKXF]] induced by the Euclidean norm.\n\nExample: $A = \\operatorname{diag}(1, 0)$ 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.\n\n## Explanations\n\n### L1 human\n\nAdding two symmetric matrices cannot move the eigenvalues in an arbitrary way.\nEach eigenvalue of the sum is bounded by sums of eigenvalues of the two parts.\nIn particular, a small symmetric perturbation moves each eigenvalue by at most the size of the perturbation.\n\n### L2 human\n\nThe corollary $\\abs{\\lambda_k(A + B) - \\lambda_k(A)} \\le \\norm{B}_2$ says that the eigenvalues of Hermitian matrices are stable under perturbation.\nFor non-Hermitian matrices this fails, and bounds such as the [[Bauer–Fike theorem|AK9J4RNZ51]] are needed instead.\nWeyl's inequalities are among the inequalities in Horn's problem on the eigenvalues of a sum of Hermitian matrices.\n",
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