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  "body": "\n## Definition\n\nLet $V$ be a vector space over $\\C$.\nLet $h \\colon V \\times V \\to \\C$ be a [[sesquilinear form|054YEQ0Y8M]], linear in the first argument and conjugate-linear in the second.\nThe form $h$ is a *Hermitian form* on $V$ when\n$$h(v, u) = \\overline{h(u, v)} \\qquad \\text{for all } u, v \\in V,$$\nwhere the bar is the [[complex conjugate|0FTH2E9TAS]].\n\n## Remarks\n\nExample: on $\\C^2$, let $h(x, y) = x_1 \\overline{y_1} + 2 x_2 \\overline{y_2}$.\nIt is sesquilinear, and $h(y, x) = y_1 \\overline{x_1} + 2 y_2 \\overline{x_2} = \\overline{h(x, y)}$, so it is Hermitian.\nNon-example: $h(x, y) = x_1 \\overline{y_2}$ on $\\C^2$ is sesquilinear but not Hermitian, because $h(e_1, e_2) = 1$ while $h(e_2, e_1) = 0$.\nTaking $u = v$ gives $h(v, v) = \\overline{h(v, v)}$, so $h(v, v)$ is real for every $v$.\nOn $\\C^n$, every Hermitian form is $h(x, y) = y^* M x$ for one [[Hermitian matrix|0P17NGQQMQ]] $M$, that is, $M^* = M$.\nSome authors take the form conjugate-linear in the first argument instead; the condition $h(v, u) = \\overline{h(u, v)}$ is the same.\n",
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