± ± 18.03 Muddy Card responses, May 10, 2006 1. “So what ‘good’ are exponential matrices? It seems to me that they don’t allow us to skip any steps: it looks like you still have to calculate eigenvalues and then eigenvectors, and then use those calculations to construct the fundamental matrix and the inverse fundamental matrix at t=0. So how do they help us, other than providing a cute analogy to what we did earlier in the course?” You know, the line between expressions of unity and coherence of a subject and cute analogies is a fuzzy one. The exponential function is a unifying theme of the course, and this is one more manifestation of it. I suggested that the expression u ( t ) = e At u (0) is convenient. A deeper reason to like the matrix exponential is that it lets you visualize the eﬀect of the diﬀerential equation on the entire plane, all at once. Think of an unstable spiral: as time progresses, the points on a circle centered at the origin rotate and move closer to the origin. This
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This note was uploaded on 01/31/2011 for the course MAT 17A taught by Professor Staff during the Winter '08 term at UC Davis.