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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^n \\times \\R$ be open, let $N \\ge 1$, and let $A_1, \\ldots, A_n \\colon \\Omega \\to \\R^{N \\times N}$ be continuous.\nThe first-order system\n$$\\partial_t u + \\sum_{j=1}^{n} A_j(x, t)\\, \\partial_{x_j} u = 0$$\nfor $u \\colon \\Omega \\to \\R^N$ is *strictly hyperbolic* if for every $(x, t) \\in \\Omega$ and every $\\xi \\in \\R^n \\setminus \\set{0}$, the matrix $A(x, t, \\xi) = \\sum_{j=1}^{n} \\xi_j A_j(x, t)$ has $N$ real and distinct [[eigenvalues|0BQAA2BK3Q]].\n\nLet $P u = 0$ be a scalar linear equation of order $m$ in the variables $(x, t)$, with [[principal symbol|F31T650DKY]] $p_m(x, t; \\xi, \\tau)$.\nIt is *strictly hyperbolic* with respect to $t$ if, for every $(x, t)$ and every $\\xi \\ne 0$, the polynomial $\\tau \\mapsto p_m(x, t; \\xi, \\tau)$ has $m$ real and distinct roots.\n\n## Remarks\n\nExample: the [[hyperbolic|92C3S03AQZ]] system $u_t + \\begin{pmatrix} 0 \u0026 -1 \\\\ -c^2 \u0026 0 \\end{pmatrix} u_x = 0$ with $c \u003e 0$, obtained from $w_{tt} = c^2 w_{xx}$ by setting $u = (w_x, w_t)$, has eigenvalues $\\pm c$, which are real and distinct.\nFor the wave operator $\\partial_t^2 - c^2 \\Delta$, the principal symbol is $-\\tau^2 + c^2 \\abs{\\xi}^2$, with the two distinct real roots $\\tau = \\pm c\\abs{\\xi}$ when $\\xi \\ne 0$.\nNon-example: the system $u_t + \\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix} u_x = 0$ has the double eigenvalue $1$, so it is not strictly hyperbolic; it is not even diagonalizable.\nStrict hyperbolicity implies that $A(x, t, \\xi)$ is diagonalizable over $\\R$, so a strictly hyperbolic system is hyperbolic.\nThe converse fails: $u_t + I u_x = 0$ with $I$ the $2 \\times 2$ identity is hyperbolic but not strictly hyperbolic.\n",
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