@matharium.official: A 2x2 matrix A turns the plane into a flow: every point moves with velocity x' = Ax. Whatever four numbers go into A, the flow is one of nine kinds. Points sink into the centre (stable node, star), shoot out of it (unstable node, degenerate node), swirl in or out (spirals), circle forever (centre), come in along one line and leave along another (saddle), or settle onto a whole line of resting points. The kind is decided by the eigenvalues of A, the roots of X½ - (tr A) + det A = 0, so only the trace and the determinant matter. det A < 0 gives a saddle. Above the parabola det A = (tr A) 4 the roots are complex: spirals, or a centre when tr A = O. Below it they are real: nodes. On it sit the star and the degenerate node, and det A = 0 gives the line of fixed points. At the end each flow slides to its own (trA, det A); the stable spiral sits at tr A = -1, det A = 2.65, where A = -0.5 ‡ 1.55i. Sound on. Follow @matharium for more. #math #matrices #differentialequations

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Friday 30 October 2026 04:05:10 GMT