372 8. MODULES k D min f m;n g , write A D diag .d 1 ;d 2 ;:::;d k / if A is diagonal and a i;i D d i for 1 ± i ± k . Proposition 8.4.6. Let A be an m –by– n matrix over R . Then there exist invertible matrices P 2 Mat m .R/ and Q 2 Mat n .R/ such that PAQ D diag .d 1 ;d 2 ;:::;d s ;0;:::;0/ , where d i divides d j for i ± j . The matrix PAQ D diag .d 1 ;d 2 ;:::;d s ;0;:::;0/ , where d i divides d j for i ± j is called the Smith normal form of A . 1 Diagonalization of the matrix A is 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 arbitrary principal ideal domain. However, we also want to pay particular attention to the case that R is a Euclidean domain, for two reasons. First, in appli-cations we will be interested exclusively in the case that R is Euclidean. Second, if R is Euclidean, Gaussian elimination is a constructive process, assuming that Euclidean division with remainder is constructive. (For a
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.