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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $\\lambda = (\\lambda_1, \\ldots, \\lambda_n)$ and $d = (d_1, \\ldots, d_n)$ be vectors in $\\R^n$.\nThen the following are equivalent.\n\n1. There is an $n \\times n$ [[Hermitian matrix|0P17NGQQMQ]] with [[eigenvalues|0BQAA2BK3Q]] $\\lambda_1, \\ldots, \\lambda_n$, counted with multiplicity, and diagonal entries $d_1, \\ldots, d_n$.\n2. The vector $d$ is [[majorized|5K7VT2VKZT]] by $\\lambda$, that is, $d \\prec \\lambda$.\n\n## Remarks\n\nSchur proved that the condition is necessary, and Horn proved that it is sufficient.\nFor necessity, write $A = U \\Lambda U^*$ with $U$ [[unitary|0AR68JTMDT]] and $\\Lambda$ diagonal; then $d_i = \\sum_j \\abs{u_{ij}}^2 \\lambda_j$.\nThe matrix $(\\abs{u_{ij}}^2)$ is [[doubly stochastic|JJA90EB2PS]], so $d \\prec \\lambda$ by the Hardy–Littlewood–Pólya characterisation of majorization.\nThe converse is proved by constructing the matrix, for example by induction on $n$.\nExample: $\\lambda = (2, 0)$ and $d = (1, 1)$ satisfy $d \\prec \\lambda$, and $\\begin{pmatrix} 1 \u0026 1 \\\\ 1 \u0026 1 \\end{pmatrix}$ has eigenvalues $2, 0$ and diagonal $1, 1$.\nThe diagonal $(3, -1)$ is impossible for the eigenvalues $(2, 0)$, since $3 \u003e 2$.\nThe same statement holds with real symmetric matrices in place of Hermitian matrices.\n\n## Explanations\n\n### L1 human\n\nThe diagonal entries of a Hermitian matrix are averages of its eigenvalues.\nSo they are less spread out than the eigenvalues, in the precise sense of majorization.\nThe theorem says that this is the only restriction: every diagonal that is less spread out than the eigenvalues actually occurs.\n\n### L2 human\n\nThe case $k = 1$ of the majorization gives $\\max_i d_i \\le \\max_j \\lambda_j$, a special case of the Rayleigh quotient bound.\nThe theorem identifies the set of diagonals of the matrices $U \\Lambda U^*$, $U$ unitary, with the permutohedron, the convex hull of the permutations of $\\lambda$.\nIn this form it is a special case of Kostant's convexity theorem and of the convexity theorem of Atiyah and of Guillemin and Sternberg for moment maps.\nIn frame theory it is used to build frames with prescribed norms and frame operator.\n",
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