Make a variable substitution. If you have a homogeneous linear system of ODEs , where A has a basis of eigenvectors and corresponding eigenvalues , then make a substitution in terms of a new vector variable y, the coordinate vector of x with respect to . Then,
Hence, the system becomes,
By the linear independence of , we may equate the (non-constant, but still scalar) coefficients of to obtain
for each i. In other words, we have now decoupled the system into n totally independent first order ODEs, in terms of the coordinates of y. These have the usual solutions:
for some constant . Thus, changing back,
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