Recently came across to solve system of linear differential equation with the matrix method, I have learnt that the eigenvectors forms part of the complementary function.

A matrix method such as :

$\frac{dy}{dt}=5x+7y$

$\frac{dx}{dt}=9x+4y$

which then we replace a vector x =(x,y) and a matrix M with entries (57,94)

Is there any reason for this (such as the one in ordinary Differential equation is due to the fact that the exponential is the eigenfunction if the differential which also exist as a matrix method, but why the eigenvectors in the front of it?)

A matrix method such as :

which then we replace a vector x =(x,y) and a matrix M with entries (57,94)

Is there any reason for this (such as the one in ordinary Differential equation is due to the fact that the exponential is the eigenfunction if the differential which also exist as a matrix method, but why the eigenvectors in the front of it?)