The Chinese Remainder Theorem gives a way to compute matrix exponentials.
Indeed, let A be a complex square matrix, put B:=mathbbC[A]. This is a Banach algebra, and also a mathbbC[X]-algebra (X being an indeterminate). Let S be the set of eigenvalues of A, mu=prodsinS(X−s)m(s)
the minimal polynomial of A, and identify B to mathbbC[X]/(mu).
The Chinese Remainder Theorem says that the canonical mathbbC[X]-algebra morphism Phi:BtoC:=prodsinSmathbbC[X]/(X−s)m(s)
is bijective.
Computing exponentials in C is trivial, so the only missing piece in our puzzle is the explicit inversion of Phi.
Fix s in S and let es be the element of C which has a one at the s place and zeros elsewhere. It suffices to compute Phi−1(es). This element will be of the form f=fracmu(X−s)m(s)gmboxmodmu
with f,ginmathbbC[X], the only requirement being gequivfrac(X−s)m(s)mumboxmod(X−s)m(s)
(the congruence taking place in the ring of rational fractions defined at s). So g is given by Taylor's Formula.
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