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  "body": "\n## Definition\n\nLet $A$ be a [[Hermitian|0P17NGQQMQ]] $n \\times n$ complex matrix, and let $x \\in \\C^n$ with $x \\ne 0$.\nThe *Rayleigh quotient* of $A$ and $x$ is the real number\n$$R(A, x) = \\frac{x^* A x}{x^* x}.$$\n\n## Remarks\n\nFor each $x \\ne 0$, $\\lambda_{\\min} \\le R(A, x) \\le \\lambda_{\\max}$, where $\\lambda_{\\min}$ and $\\lambda_{\\max}$ are the least and the greatest [[eigenvalues|0BQAA2BK3Q]] of $A$.\nIf $x$ is an eigenvector for $\\lambda$, then $R(A, x) = \\lambda$.\nSo $\\lambda_{\\max} = \\max_{x \\ne 0} R(A, x)$ and $\\lambda_{\\min} = \\min_{x \\ne 0} R(A, x)$.\n\n## Explanations\n\n### L1 human\n\nThe Rayleigh quotient measures how much a symmetric matrix stretches a given direction, on average.\nIts largest value over all directions is the largest eigenvalue.\nThis gives a way to estimate eigenvalues without solving the characteristic equation.\n",
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