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Solving Linear Differential Equations via the Matrix Exponential exp(At)

In systems theory and linear algebra, first-order constant-coefficient differential equations $\dot{\mathbf{u}} = A\mathbf{u}$ exhibit solutions defined by the matrix exponential $e^{At}$. This concept parallels discrete power sequences through diagonalization via eigenvector matrices ($S$), where stability and steady states are strictly determined by the real parts of eigenvalues lying in the left half-plane.

Dr. Harry Seldon Video

Solving Linear Differential Equations via the Matrix Exponential exp(At)

Solving Systems of Differential Equations with the Matrix Exponential e^{At}