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?