{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

College Algebra Exam Review 362

# College Algebra Exam Review 362 - 372 8 MODULES k D minfm...

This preview shows page 1. Sign up to view the full content.

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
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Ask a homework question - tutors are online