5.4 Calculation of Invariant Factors

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In this section, R continues to be a principal ideal domain. If A,B 2

Mm,n(R), we say that A, B are Smith equivalent (and write A _S B) if there

exist invertible matrices P 2 Mn(R), Q 2 Mm(R) such that B = QAP. We

say that the matrix A = [aij ] 2 Mm,n(R) is in Smith canonical form if

(i) i 6= j implies that aij = 0.

(ii) There exists r such that a11, a22, . . . , arr 6= 0, and all ass = 0, if s > r.

(iii) If we set ai = aii, i = 1, 2, . . . , r, then ai|ai+1, i = 1, 2, . . . , r − 1.

Theorem 5.4.1 If A 2 Mm,n(R), then A is Smith equivalent to a matrix

in Smith canonical form.

We now discuss the relationship of the above with the structure of finitely

generated R-modules, where R is a principal ideal domain. Thus, Let M =

Rhx1, x2, . . . , xni; if F = Rhe1, e2, . . . , eni is free with basis {e1, e2, . . . , en},

then there is a unique homomorphism _ : F ! M, with ei 7! xi, i =

1, 2, . . . , n. Let K = ker _; thus K is a free R module of F with generators

fj =

Xn

i=1

ajiei, j = 1, 2, . . . , m.

In other words, we have a presentation of the R-module M in much the

same way as one has presentations of groups:

M _= Rhe1, . . . , en|

X

aijej = 0, i = 1, . . . ,mi.

The matrix A = [aij ] 2 Mmn(R) is called a relations matrix for the module

M.

Conversely, given a matrix A = [aij ] 2 Mmn(R), we define a module

MA = Rhe1, . . . , en|

X

aijej = 0, i = 1, . . . ,mi.

Therefore, any finitely generated module over the p.i.d. R is isomorphic

with MA for some matrix A with coefficients in R.

Proposition 5.4.2 Let A,B 2 Mmn(R), and assume that A and B are

Smith equivalent. Then MA

_= MB.

120 CHAPTER 5. MODULE THEORY

In, particular, when M _= MA and when D is Smith equivalent to A and

is in Smith canonical form, the structure of M is obtained as follows:

Theorem 5.4.3 Let M _= MA and assume that A is equivalent to D = [dij ],

where S is in Smith canonical form. Set di = dii, i = 1, . . . , min {m, n},

and if m < n, set dm+1, . . . , dn = 0. Then

M _= R/Rd1 _ R/Rd2 _ · · ·R/Rdn.

Note that if d1, d2, . . . , dr are non-zero non-units, then d1, d2, . . . , dr are

precisely the invariant factors of M.

Corollary 5.4.3.1 Let A 2 Mmn(R) and assume that D = [dij ],D0 = [d0

ij ]

are Smith equivalent to A and are in Smith canonical form. Then dij _ d0

ij

(associates). Thus, the “Smith canonical form” of a matrix A 2 Mmn(R) is

unique up to associates.

The following result is sometimes convenient for “small” relations matrices.

Let A = [aij ] 2 Mmn(R). An i-rowed minor of A is simply the

determinant of an i×i submatrix of A. Say that A is of determinantal rank

r if there exists a non-zero r-rowed minor, but every (r + 1)-rowed minor

is 0. Let _ = _i(A) be the greatest common divisor of all of the i-rowed

minors of A. Note that _i|_i+1, i = 1, 2, . . . , r − 1. We have

Theorem 5.4.4 Assume that A has determinantal rank r, and that _1,_2, . . . ,_r

are as above. Set

d1 = _1, d2 = _2_−1

1 , . . . , dr = _r_−1

r−1.

Then d1, d2, . . . , dr are the non-zero invariant factors of A.

Exercises

1. Suppose we have the finitely generated abelian group

G = he1, . . . , en|

X

aijej = 0 i,

where the relations matrix A = [aij ] is a square matrix. Show that G

is finite if and only if det (A) 6= 0, in which case |G| = |det (A)|.

5.4. CALCULATION OF INVARIANT FACTORS 121

2. Compute the structure of the abelian group

he1, . . . , en|

X

aijej = 0 i,

given that

(a)

A =

2

4

6 2 3

2 3 −4

−3 3 1

3

5.

(b)

A =

2

4

2 −1 0

−1 2 −1

0 −1 2

3

5.

(c)

A =

2

4

2 −1 0

−1 2 −1

0 −2 2

3

5.

(d)

A =

2

664

2 −1 0 0

−1 2 −1 −1

0 −1 2 0

0 −1 0 2

3

775

.

(e)

A =

2

666666664

2 −1 0 . . . .

−1 2 −1 . . . .

0 −1 2 . . . .

. . . . . . .

. . . . 2 −1 0

. . . . −1 2 −1

. . . . 0 −1 2

3

777777775

.

3. Let A 2 Mn(R). Show that A _S At.

122 CHAPTER 5. MODULE THEORY

4. Suppose that

M = MA = Rhe1, . . . , en|

X

aijej = 0, i = 1, . . . ,mi.

If PAQ = D is in Smith canonical form, show how to obtain a generating

set for T(M), the torsion submodule of M, as R-linear combinations

of the generators e1, e2, . . . en of MA.

5. Let R be a p.i.d. and let A,B 2 Mn(R), where B is an invertible

matrix. If M is the kn × kn block matrix

M =

2

66664

A 0 . . 0

B A . . 0

0 B . . .

. . . . .

0 0 . B A

3

77775

,

show that M and Ak have the same non-trivial (i.e., non-unit) invariant

factors. Put differently, show that M and

M0 =

2

66664

I 0 . . 0

0 I . . 0

0 0 . . .

. . . . .

0 0 . 0 Ak

3

77775

,

are Smith equivalent.

5.5. APPLICATION TO A SINGLE LINEAR TRANSFORMATION 123

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