Sunday, 9 January 2011

examples - Applications of the Chinese remainder theorem

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(Xs)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]/(Xs)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 Phi1(es). This element will be of the form f=fracmu(Xs)m(s)gmboxmodmu

with f,ginmathbbC[X], the only requirement being gequivfrac(Xs)m(s)mumboxmod(Xs)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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