3728. MODULESkDminfm; ng, writeADdiag.d1; d2; : : : ; dk/ifAis diagonal andai;iDdifor1ik.Proposition 8.4.6.LetAbe anm–by–nmatrix overR. Then there existinvertible matricesP2Matm.R/andQ2Matn.R/such thatPAQDdiag.d1; d2; : : : ; ds; 0; : : : ; 0/, wheredidividesdjforij.The matrixPAQDdiag.d1; d2; : : : ; ds; 0; : : : ; 0/, wheredidividesdjforijis called theSmith normal formofA.1Diagonalization of the matrixAis accomplished by a version of Gauss-ian elimination (row and column reduction). For the sake of completeness,we will discuss the diagonalization process for matrices over an arbitraryprincipal ideal domain. However, we also want to pay particular attentionto the case thatRis a Euclidean domain, for two reasons. First, in appli-cations we will be interested exclusively in the case thatRis Euclidean.Second, ifRis Euclidean, Gaussian elimination is a constructive process,assuming that Euclidean division with remainder is constructive. (For a
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