snf_lin-eqns - Smith Normal Form and Integer solutions to...

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Smith Normal Form and Integer solutions to linear equations Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] 26 August, 2007 (at 21:07 ) In this tract, Gcd(0 ,..., 0) is zero; this, since every integer divides zero. All matrices (and vectors) are integer-valued , 1 unless specified otherwise . So a square matrix R is invertible 1 IFF Det( R ) ∈ {± 1 } . For two, say, 3 × 5 matrices A and B , write: A r B , read “ A is row equivalent to B ”; A c B , read “ A is column equiv. to B ”; A r c ∼∼ B , read “ A is rowcol equivalent to B ”; if there exist invertible matrices 3 × 3 R and 5 × 5 C such that, respectively, R A = B ; A C = B ; R A C = B . Smith Normal Form An r × c matrix G δ 1 0 δ 2 0 . . . . . . δ m 0 ( Here, m := Min( r , c ). In this example, c equals r + 1. All unshown entries are zero. ) 5 : is in Smith form if its only non-zero entries –the pivot-values – are on its main-diagonal. (Zeros are allowed on the main-diagonal). Smith form requires that the matrix be integer-valued, and that all non- zero values on the diagonal occur before the zeros. Let π = π ( G ) denote the number of pivots (non-zero values). Our r × c matrix G is in Smith normal form δ 1 0 . . . . . . δ π . . . 0 ... ... 0 ... 0 ( In this example, m = r and π = r - 1. Thus δ m is necessarily 0. ) 6 : 1 More generally, our matrices’ entries come from a euclidean domain , ED . An invertible R has Det( R ) Units( ED ). For row operations, we may: Add any ED -multiple of a row to another; Multiply a row by any ED - unit . Ditto for column-ops. if, in addition, the pivots-values are positive and δ 1 •| ... •| δ π - 1 •| δ π •| ··· •| δ m . 7 : ( This divisibility-condition forces zeros on the diagonal to occur last. ) Elementary row operations. Applied to an r × c matrix Γ, the (elementary) row-operations 1 are a : Exchanging two rows. b : Adding a Z -multiple of one row to another. c : Multiplying a row by 1. Applying a row-op to Γ produces E Γ, where E is an r × r elementary matrix . Notice that Det( E ) is ± 1, since we are allowed to multiply a row only 1 by 1. Analogously, there are the column operations . Applying a col-op to Γ produces Γ b E , where b E is an c × c elementary matrix. Applying j –many row-ops and k –many col-ops to Γ produces a matrix G := R Γ C , where R := E j ··· E 2 E 1 and C := b E 1 b E 2 ··· b E k . This R is an integer matrix with Det( R ) ∈ {± 1 } . Ditto C . These are the bookkeeping matrices; our R keeps track of the (cumulative) row-ops, and C keeps track of the col-ops. Converting Γ to G via elem. row-ops and col-ops manifests
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This note was uploaded on 01/26/2012 for the course MAC 3472 taught by Professor Jury during the Fall '07 term at University of Florida.

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snf_lin-eqns - Smith Normal Form and Integer solutions to...

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