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  "body": "\n## Statement\n\nLet $1 \\le m \\le n$, and let $A$ be an $n \\times n$ [[Hermitian matrix|0P17NGQQMQ]] with [[eigenvalues|0BQAA2BK3Q]] $\\alpha_1 \\ge \\dots \\ge \\alpha_n$.\nLet $P$ be an $n \\times m$ complex matrix with $P^* P = I_m$, and let $B = P^* A P$, with eigenvalues $\\beta_1 \\ge \\dots \\ge \\beta_m$.\nThen, for $j = 1, \\dots, m$,\n$$\\alpha_j \\ge \\beta_j \\ge \\alpha_{j + n - m}.$$\nIn particular, if $B$ is obtained from $A$ by deleting row $i$ and column $i$ for some $i$, then\n$$\\alpha_1 \\ge \\beta_1 \\ge \\alpha_2 \\ge \\beta_2 \\ge \\dots \\ge \\alpha_{n-1} \\ge \\beta_{n-1} \\ge \\alpha_n.$$\n\n## Remarks\n\nBoth inequalities follow from the [[Courant–Fischer min-max theorem|CJK2HS2508]].\nA subspace $W$ of $\\C^m$ of dimension $j$ maps to the subspace $PW$ of $\\C^n$ of dimension $j$, and $P$ preserves the Rayleigh quotient: $R_B(w) = R_A(Pw)$.\nSo the max-min for $\\beta_j$ runs over fewer subspaces than the one for $\\alpha_j$, which gives $\\beta_j \\le \\alpha_j$.\nThe min-max form gives the other inequality.\n\nExample: $A = \\begin{pmatrix} 2 \u0026 1 \\\\ 1 \u0026 2 \\end{pmatrix}$ has eigenvalues $3$ and $1$.\nDeleting the second row and column leaves $(2)$, and indeed $3 \\ge 2 \\ge 1$.\n\n## Explanations\n\n### L1 human\n\nRemove one row and the matching column from a symmetric matrix.\nThe eigenvalues of the smaller matrix fit between those of the larger one, one in each gap.\nSo a principal submatrix cannot have eigenvalues far from those of the whole matrix.\n\n### L2 human\n\nInterlacing is used to bound eigenvalues of graphs: the adjacency matrix of an induced subgraph is a principal submatrix.\nApplied repeatedly, it compares the eigenvalues of $A$ with those of each of its principal submatrices.\nThe converse holds too: any two interlacing sequences arise from a Hermitian matrix and a principal submatrix of size one less.\n",
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