eigenvectors is used to diagonalized a matrix
if A is diagonalizable so A = P (given by eigenvectors (A) ) * D (given by eigenvalues(A) ) *P^-1
"and systems of lienar diff eqn"
is it something like ?
y'(t) = a*x(t) +b*y(t) +c
x'(t) = d*x(t) +e*y(t)+f .
if so the system can be seen as matrix you have to diagonalized one and use some properties to solve it (would have to check my lesson

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if you have any question feel free to asj