18.03 Lecture Notes - Lecture 36: Matrix Exponential, Linear Combination, Phase Portrait
Document Summary
The matrix exponential: initial value problems: definition of e^{at, computation of e^{at, uncoupled example, defective example, exponential law. [1] recall from day one: (a) x" = rx with initial condition x(0) = 1 has solution x = e^{rt} . x" = rx with any initial condition has solution x = e^{rt} x(0) . Later, we decided to *define* e^{it} as the solution to (b) x" = ix with initial condition x(0) = 1 . Following euler, a solution is given by cos t + i sin t , so we found that e^{it} = cos(t) + i sin(t) (c) now we are studying u" = a u . The solution to u" = au with initial condition u(0) is u = e^{at}u(0). Note that the initial value u(0) is a vector, and u(t) is a vector valued function. So the expression e^{at} must denote a matrix, or rather a matrix valued function.
