# Nonlinearities arising from linear equations

I have sources that tell me that the Bloch Equations, which describe the magnetization vector in nuclear magnetic resonance, are nonlinear. In vector form, without relaxation, they are: $$\partial_t \vec{M}(t) = \gamma \vec{M}(t) \times \vec{B}$$

Nothing seems nonlinear to me. Am I missing something from tensor calculus？

Edit: the above equation can be expressed in matrix from, and is hence a linear equation (see link above). Still, papers that I read about NMR lead me to believe that there is some fact about the Bloch equations that can cause a nonlinear evolution. Is it possible for the path of a state vector following this equation to be nonlinear, in any sense of the word?

• What are your sources? Often a constant term appears as well. This could be a source of non-linearity. – plan Dec 26 '17 at 5:38