> A := Matrix(2,2,[1,2,3,4]); > A; [1 2] [3 4] > SmithForm(A); [1 0] [0 2] [ 1 0] [-1 1] [-1 2] [ 1 -1]As you can see, computed the Smith form, which is . What are the other two matrices it output? To see what any command does, type the command by itself with no arguments followed by a semicolon.

> SmithForm; Intrinsic 'SmithForm' Signatures: (<Mtrx> X) -> Mtrx, AlgMatElt, AlgMatElt [ k: RngIntElt, NormType: MonStgElt, Partial: BoolElt, RightInverse: BoolElt ] The smith form S of X, together with unimodular matrices P and Q such that P * X * Q = S.As you can see,

> S, P, Q := SmithForm(A); > S; [1 0] [0 2] > P; [ 1 0] [-1 1] > Q; [-1 2] [ 1 -1] > P*A*Q; [1 0] [0 2]Next, let's test the limits. We make a integer matrix with entries between 0 and , and compute its Smith normal form.

> A := Matrix(10,10,[Random(100) : i in [1..100]]); > time B := SmithForm(A); Time: 0.000Let's print the first row of , the first and last row of , and the diagonal of :

> A[1]; ( 4 48 84 3 58 61 53 26 9 5) > B[1]; (1 0 0 0 0 0 0 0 0 0) > B[10]; (0 0 0 0 0 0 0 0 0 51805501538039733) > [B[i,i] : i in [1..10]]; [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 51805501538039733 ]Let's see how big we have to make in order to slow down . These timings below are on a 1.6Ghz Pentium 4-M laptop running Magma V2.11 under VMware Linux. I tried exactly the same computation running Magma V2.17 natively under Windows XP on the same machine, and it takes

> n := 50; A := Matrix(n,n,[Random(100) : i in [1..n^2]]); > time B := SmithForm(A); Time: 0.050 > n := 100; A := Matrix(n,n,[Random(100) : i in [1..n^2]]); > time B := SmithForm(A); Time: 0.800 > n := 150; A := Matrix(n,n,[Random(100) : i in [1..n^2]]); > time B := SmithForm(A); Time: 4.900 > n := 200; A := Matrix(n,n,[Random(100) : i in [1..n^2]]); > time B := SmithForm(A); Time: 19.160

can also work with finitely generated abelian groups.

> G := AbelianGroup([3,5,18]); > G; Abelian Group isomorphic to Z/3 + Z/90 Defined on 3 generators Relations: 3*G.1 = 0 5*G.2 = 0 18*G.3 = 0 > #G; 270 > H := sub<G | [G.1+G.2]>; > #H; 15 > G/H; Abelian Group isomorphic to Z/18

William Stein 2004-05-06