The Matrix Exponential: a Flow Whose Area Grows like e to the Trace
An interactive live film: a ring of initial states flows upward in time under e^{tA}, turning and swelling; det e^{tA} = e^{t tr A}. Choose the trace so the area stays constant.
Narration
A linear differential equation, x prime equals A x, moves every point of the plane. Its solution is x of t equals the matrix exponential of t A, times the starting point.
The exponential is the same power series as for numbers: the identity, plus t A, plus t A squared over two, and so on. It converges for every square matrix.
The rest of the narration is in the book's Companion.
More from Chapter 6
Interactive1:32Eigenvectors: the Directions a Matrix Does Not TurnChapter 6
Preview1:42The Determinant as a Volume Factor, with a SignChapter 6
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Preview1:46Power Method: A Markov Chain Finds Its EigenvectorChapter 6
Preview1:44Diagonalization: Change Basis, Stretch, Change BackChapter 6