Colloquium publications.

108

THE CAMBRIDGE COLLOQUIUM.

Normal Form of Ek
14. Let the rank of Ek (k = 0, 1, **, n) be denoted by rk.
By the theory of matrices whose elements are integers (cf. ~ 49,
Chap. 1) there exist square matrices Ck_- and Dk with integer
elements and determinants 4t 1, of ak-i and ak rows respectively, such that
(5) Ck-1-'1Ek'Dk = Ek
where Ek* is a matrix of ak-i rows and ak columns all the elements
of which are zero except the first rk elements of the main diagonal,
which are the invariant factors of Ek. We shall denote the elements of the main diagonal of Ek* by djk. and understand that
djk = 0 if j > rk.
Equation (5) is equivalent to
(6) Ek'Dk = Ck-1'Ek*,
and (6) may be regarded as a set of ak equations of the form,
(7) Ek. xij = dj yj i
x2j dj Y2j
^o^ I djk I2j
Xaki d? Yak-li
in which xij, x2j, *, Xai are the elements of the jth column of
Dk and yij, y2j, ''', y,-, those of the jth column of Ck-. By
~ 11, this means that the jth column of Dk represents an oriented
complex the boundary of which covers the oriented complex
represented by the jth column of Ck-i a number of times equal
to the jth element of the main diagonal of Ek*.
15. Since the last ak - rk columns of Ek* are composed entirely
of zeros, the last ak- rk columns of Dk represent a complete
set of k-circuits or sets of k-circuits. As in ~ 12, Chap. III,
these columns may be modified without affecting the equations
(6) so that each column represents a single k-circuit. Let this
be done for all values of k, k = 0, 1, 2,.., n.