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  "body": "\n## Statement\n\nLet $A$ be an $n \\times n$ [[Hermitian matrix|0P17NGQQMQ]] with [[eigenvalues|0BQAA2BK3Q]] $\\lambda_1 \\ge \\lambda_2 \\ge \\dots \\ge \\lambda_n$, listed with multiplicity.\nFor $x \\in \\C^n$ with $x \\ne 0$, let $R_A(x) = \\dfrac{x^* A x}{x^* x}$ be the [[Rayleigh quotient|06WHP8SE98]].\nThen, for each $k = 1, \\dots, n$,\n$$\\lambda_k = \\max_{\\dim U = k} \\; \\min_{x \\in U \\setminus \\set{0}} R_A(x) = \\min_{\\dim U = n-k+1} \\; \\max_{x \\in U \\setminus \\set{0}} R_A(x),$$\nwhere $U$ runs over the [[linear subspaces|00YC1D652K]] of $\\C^n$ of the given [[dimension|03GSZ8RTZJ]].\n\n## Remarks\n\nLet $u_1, \\dots, u_n$ be an orthonormal basis of eigenvectors with $A u_i = \\lambda_i u_i$.\nOn the span of $u_1, \\dots, u_k$, the Rayleigh quotient is at least $\\lambda_k$, so the max-min is at least $\\lambda_k$.\nEvery subspace $U$ of dimension $k$ meets the span of $u_k, \\dots, u_n$ in a nonzero vector, by counting dimensions, and there $R_A \\le \\lambda_k$.\nSo the max-min is exactly $\\lambda_k$; the min-max form is proved in the same way.\n\nFor $k = 1$ it gives $\\lambda_1 = \\max_{x \\ne 0} R_A(x)$, and for $k = n$ it gives $\\lambda_n = \\min_{x \\ne 0} R_A(x)$.\nFor $A = \\operatorname{diag}(3, 1)$, $R_A(x) = (3x_1^2 + x_2^2)/(x_1^2 + x_2^2)$ for real $x$, which takes all values in $[1, 3]$.\n\nThe Hermitian hypothesis is needed: for a non-Hermitian matrix the eigenvalues need not be real, and $R_A$ need not be real either.\n\n## Explanations\n\n### L1 human\n\nThe eigenvalues of a symmetric matrix can be found by an optimisation, without solving any polynomial equation.\nThe largest eigenvalue is the largest value of the Rayleigh quotient.\nThe second one is the best value one can guarantee on every plane, and so on.\n\n### L2 human\n\nThe theorem turns questions about eigenvalues into questions about subspaces.\nIt gives the [[Cauchy interlacing theorem|8QQPVNR80F]] and [[Weyl's inequality|RACSNQXDQV]] in a few lines.\nThe real symmetric case is the same, with $\\R^n$ and $x^T A x$.\nA version holds for compact self-adjoint operators on Hilbert spaces.\n",
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