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  "body": "\n## Summary\n\nA Hermitian form on a complex vector space is a [[sesquilinear form|054YEQ0Y8M]] $h$ with $h(v, u) = \\overline{h(u, v)}$ for all vectors $u, v$.\nIt is the complex analogue of a [[symmetric bilinear form|0AXZNE5XNH]].\nThe value $h(v, v)$ is always real, so Hermitian forms can be positive, negative or indefinite.\nComplex [[inner products|0NB0AWGJBW]] are exactly the positive definite Hermitian forms.\n\n## Explanations\n\n### L1 human\n\nOver the real numbers, symmetric bilinear forms such as the dot product measure squared lengths.\nOver the complex numbers, one has to conjugate one argument to keep $h(v, v)$ real; this gives a Hermitian form.\nOn $\\C^n$, the Hermitian forms are exactly $h(x, y) = y^* M x$ with $M$ a [[Hermitian matrix|0P17NGQQMQ]].\nFor example, $h(x, y) = x_1 \\overline{y_1} - x_2 \\overline{y_2}$ on $\\C^2$ is Hermitian, with $h(v, v) = \\abs{v_1}^2 - \\abs{v_2}^2$.\n",
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