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Linear Algebra, Optimisation and Numerical Methods / Eigenvalues, Diagonalisation, and Dynamical Applications

Applications: Systems of Linear Differential Equations du/dt = Au

The previous note solved the discrete recursion uk+1=Auk\mathbf{u}_{k+1}=A\mathbf{u}_k via eigenvalues. This note solves its continuous-time twin, du/dt=Aud\mathbf{u}/dt = A\mathbf{u} — a system of coupled linear ODEs, such as several mean-reverting rates or prices influencing each other continuously — by exactly the same eigenvalue method, with λk\lambda^k replaced by eλte^{\lambda t}. The note closes Topic 6 on its most practically consequential reading: the sign of the real part of each eigenvalue determines whether that mode of the system explodes, decays, or oscillates forever.
Learning objectives. By the end of this note you should be able to:
  • solve du/dt=Aud\mathbf{u}/dt=A\mathbf{u} in closed form using AA's eigendecomposition;
  • verify a proposed solution by direct differentiation;
  • classify a system's stability from the sign of Re(λ)\text{Re}(\lambda) for each eigenvalue;
  • explain the parallel structure between the discrete and continuous cases.

In this lesson

  1. Solving du/dt=Aud\mathbf{u}/dt = A\mathbf{u}
  2. Stability from the sign of Re(λ)\text{Re}(\lambda)

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