Linearisation about an equilibrium and the eigenvalue stability criterion (Q1677)

\\mathbf{u}'=A\\mathbf{u},\\qquad A=D\\mathbf{f}(\\mathbf{x}^{*});\\qquad \\operatorname{Re}(\\lambda)<0\\ \\text{for all eigenvalues } \\lambda\\ \\Rightarrow\\ \\mathbf{x}^{*}\\ \\text{asymptotically stable}