https://www.chenk.top/en/tags/linear-systems/Recent content in Linear Systems on Chen Kai BlogHugoenSun, 24 Sep 2023 09:00:00 +0000https://www.chenk.top/en/ode/06-power-series/Sun, 24 Sep 2023 09:00:00 +0000https://www.chenk.top/en/ode/06-power-series/<p><strong>One equation describes one quantity. The world is rarely that obliging.</strong> Predator and prey populations push each other up and down. Currents and voltages in an RLC network oscillate together. Chemical species in a reaction network feed into one another. The moment two unknowns share an equation, you have a <em>system</em>, and a single <span class="math-inline">$y&#39;=ay$</span> is no longer enough.</p> <p>The miracle of the linear case is this: the scalar formula <span class="math-inline">$y(t)=e^{at}y_0$</span> generalizes verbatim once you learn what <span class="math-inline">$e^{At}$</span> means for a <em>matrix</em> <span class="math-inline">$A$</span> . Linear algebra and ODEs fuse into one object — the matrix exponential — and its eigenstructure tells you everything about the long-term behavior, the geometry of the flow, and the physics of normal modes and beats.</p>