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Berkeley
MATH 54 -
Systems with constant coefficients
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University of California, Berkeley
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Systems with constant coefficientsNovember 14, 2007Suppose the entries of A are constant. Then the solutions tothe homogeneous equationx0= Axcan often be found explicitly using eigenvectors andeigenvaluesTheoremIf vi∈ Rnis an eigenvector of A with eigenvalue λi, then thevector-functionxi:= eλitviis a solution to the equation x0= Ax.ProofWe just check:x0i= (eλitvi)0= (eλit)0vi= eλitλivi= eλitAvi= A(eλitvi)= AxiMain conclusionIf A has n distinct eigenvalues λ1, . . . , λn, with correspondingeigenvectors v1, . . . , vn, then(eλ1tv1, . . . eλntvn)is a basis for the space of solutions to the equation x0=
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