I'm studying a course in dynamical systems. It's a pretty much linear algebra intensive course, and it's been a while since I did that sort of things.
In it, they say that if vector $z$ satisfies $Az=\lambda z$, then $x(t) = c e^{\lambda t} z$ is a solution to $dx/dt = Ax$.
So far so good.
Further, they claim that if there are several solutions to the equation $Az = \lambda z$, that is $A z_1 = \lambda_1 z_1, A z_2 = \lambda_2 z_2, ... , A z_p = \lambda_p z_p$, then according to the superposition principle it follows that for all $c_1, c_2, ... , c_p$ the vector function $x(t) = c_1*e^{\lambda_1 t}z_1+c_2*e^{\lambda_2 t} z_2+...+c_p*e^{\lambda_p t}z_p$ is a solution to $dx/dt=Ax$. I don't understand this reasoning. Could anybody explain why it follows from the superposition principle. Tried all day get my head around it.
Given your final expression for $x(t)$,
$\displaystyle\frac{dx}{dt}=c_1\lambda_1 e^{\lambda_1 t} z_1+ c_2\lambda_2 e^{\lambda_2 t} z_2+ ... + c_p\lambda_p e^{\lambda_p t} z_p$.
Given your immediately preceding expressions relating $A$, $z_1, \lambda_1$, $z_2, \lambda_2$ etc.,
$\displaystyle\frac{dx}{dt}=c_1 e^{\lambda_1 t} Az_1 + c_2 e^{\lambda_2 t} Az_2 + ... + c_p e^{\lambda_p t} Az_p = A(c_1 e^{\lambda_1 t} z_1 + c_2 e^{\lambda_2 t} z_2 + ... + c_p e^{\lambda_p t} z_p)=$
$=Ax$
So your superposed expression is a solution of $\displaystyle\frac{dx}{dt}=Ax$.