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All of my yearlong graduate-level Linear Algebra course notes from my professor—an algebraist/representation theorist—shows his love for the exponential map $e^A$ and the Jordan canonical form—and one main reason given by him is that they help solve linear differential equations.

But how true is this in practice?

I have little scientific computing experience.

The thing is, in my research interest, fluid dynamics, not only are the evolution equations nonlinear, often with nonconstant coefficients. Do the matrix exponential and the Jordan form help with modeling and simulation in fluid dynamics problems?

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    $\begingroup$ Krylov subspace methods for computing matrix exponential-vector products are more often used because they save significant time compared to diagonalization/Jordan form and other direct methods of computing the exponential and performing the multiplication $\endgroup$ – whpowell96 Jun 22 at 19:52
  • $\begingroup$ Fluids are often their own beast as far as algorithms but check out "exponential integrators" for the state-of-the-art in computing and utilizing matrix exponentials $\endgroup$ – whpowell96 Jun 22 at 19:56

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