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Stability of fixed points in 2D

The paradigmatic case is the stability of the origin under the linear autonomous differential equation X ˙ = A X {\displaystyle {\dot {X}}=AX} where X = {\displaystyle X={\begin{bmatrix}x\\y\end{bmatrix}}} and A {\displaystyle A} is a 2-by-2 matrix.

We would sometimes perform change-of-basis by X ′ = C X {\displaystyle X'=CX} for some invertible matrix C {\displaystyle C}, which gives X ˙ ′ = C − 1 A C X ′ {\displaystyle {\dot {X}}'=C^{-1}ACX'}. We say C − 1 A C {\displaystyle C^{-1}AC} is " A {\displaystyle A} in the new basis". Since det A = det C − 1 A C {\displaystyle \det A=\det C^{-1}AC} and tr A = tr C − 1 A C {\displaystyle \operatorname {tr} A=\operatorname {tr} C^{-1}AC}, we can classify the stability of origin using det A {\displaystyle \det A} and tr A {\displaystyle \operatorname {tr} A}, while freely using change-of-basis.