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  "body": "\n## Definition\n\nLet $\\partial_t u + \\sum_{j=1}^n \\partial_{x_j} F_j(u) = 0$ be a system of [[conservation laws|8RA7AMWSAF]], with $C^1$ fluxes $F_j \\colon \\R^m \\to \\R^m$ and unknown $u$ with values in an open set $\\mathcal{U} \\subseteq \\R^m$.\nFor $u \\in \\mathcal{U}$ and $\\xi \\in \\R^n$ let $B(u; \\xi) = \\sum_{j=1}^n \\xi_j\\, DF_j(u)$, an $m \\times m$ real matrix, where $DF_j$ is the Jacobian matrix of $F_j$.\nThe system is *hyperbolic* on $\\mathcal{U}$ if, for every $u \\in \\mathcal{U}$ and every unit vector $\\xi \\in \\R^n$, the matrix $B(u; \\xi)$ is diagonalizable over $\\R$.\nThat is, $B(u; \\xi)$ has $m$ real eigenvalues $\\lambda_1(u; \\xi) \\le \\cdots \\le \\lambda_m(u; \\xi)$ and a basis of $\\R^m$ of corresponding eigenvectors.\n\n## Remarks\n\nIt is *strictly hyperbolic* if moreover the eigenvalues are distinct, $\\lambda_1 \u003c \\cdots \u003c \\lambda_m$.\nIn one space dimension $B(u; \\xi) = \\xi\\, DF(u)$, and the condition is that $DF(u)$ is diagonalizable with real eigenvalues; these are the characteristic speeds.\nThe definition makes the system a [[hyperbolic|92C3S03AQZ]] system of first-order equations when written in the non-conservative form $u_t + \\sum_j DF_j(u)\\, u_{x_j} = 0$.\n\nExample: every scalar conservation law ($m = 1$) is hyperbolic, since $B(u; \\xi) = F'(u) \\cdot \\xi$ is a real $1 \\times 1$ matrix.\nExample: the $p$-system $v_t - w_x = 0$, $w_t + p(v)_x = 0$ with $p' \u003c 0$ has $DF = \\begin{pmatrix} 0 \u0026 -1 \\\\ p'(v) \u0026 0 \\end{pmatrix}$.\nIts eigenvalues $\\pm \\sqrt{-p'(v)}$ are real and distinct, so the system is strictly hyperbolic.\n\nNon-example: the linear system $u_t + A u_x = 0$ with $A = \\begin{pmatrix} 0 \u0026 -1 \\\\ 1 \u0026 0 \\end{pmatrix}$, so that $F(u) = Au$.\nThe eigenvalues of $A$ are $\\pm i$, which are not real, so the system is not hyperbolic; its components satisfy the Cauchy–Riemann equations, which are elliptic.\n",
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