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  "body": "\n## Summary\n\nLet $A$ be a real $2 \\times 2$ matrix with $\\det A \\neq 0$, let $\\tau = \\operatorname{tr} A$ and $\\Delta = \\det A$, and consider $x' = A x$ in the [[phase plane|05VDGH129F]].\nThe origin is a saddle if $\\Delta \u003c 0$. It is a node if $\\Delta \u003e 0$ and $\\tau^2 - 4 \\Delta \\ge 0$, and a spiral, or focus, if $\\tau \\neq 0$ and $\\tau^2 - 4 \\Delta \u003c 0$. It is a centre if $\\tau = 0$ and $\\Delta \u003e 0$.\nA node or a spiral is stable for $\\tau \u003c 0$ and unstable for $\\tau \u003e 0$.\n\n## Remarks\n\nThe eigenvalues are the roots of $\\lambda^2 - \\tau \\lambda + \\Delta = 0$. A saddle has real eigenvalues of opposite signs. A node has real eigenvalues of the same sign. A spiral or a centre has complex conjugate eigenvalues.\nThe case $\\tau^2 = 4 \\Delta$ gives a repeated eigenvalue. The origin is then a star node or a degenerate node.\nThe trace and the determinant do not change under a change of basis $A \\mapsto C^{-1} A C$. So the classification depends only on the system, not on the coordinates.\nThe [[damped harmonic oscillator|0RNXVVTJTJ]] gives a stable spiral when it is underdamped and a stable node when it is overdamped.\n\n## Explanations\n\n### L1 human\n\nFor a linear system in the plane, two numbers, the trace and the determinant, tell the shape of the phase portrait near the origin.\nFor a node, trajectories run into or out of the origin along straight directions. They turn around it for a spiral, and they go around it in closed loops for a centre.\n",
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