Stability theory
Classification of stability types

If det A = 0 {\displaystyle \det A=0}, then the rank of A {\displaystyle A} is zero or one.

If det A ≠ 0 {\displaystyle \det A\neq 0}, we first find the Jordan normal form of the matrix, to obtain a basis { v, w } {\displaystyle \{v,w\}} in which A {\displaystyle A} is one of three possible forms:

The summary is shown in the stability diagram on the right. In each case, except the case of 4 det A − (tr A) 2 = 0 {\displaystyle 4\det A-(\operatorname {tr} A)^{2}=0}, the values (tr A, det A) {\displaystyle (\operatorname {tr} A,\det A)} allows unique classification of the type of flow.

For the special case of 4 det A − (tr A) 2 = 0 {\displaystyle 4\det A-(\operatorname {tr} A)^{2}=0}, there are two cases that cannot be distinguished by (tr A, det A) {\displaystyle (\operatorname {tr} A,\det A)}. In both cases, A {\displaystyle A} has only one eigenvalue, with algebraic multiplicity 2.
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