Analysis Algorithm for Z
Table 7.1. Degree of simplices of filtration in Figure 7.11
a b
c d
ab bc
cd ad
ac abc
acd 1
1 1
1 2
2 3
4 5
bases for
C
k
and
C
k
−
1
, respectively. Relative to homogeneous bases, any rep- resentation M
k
of ∂
k
has the following basic property: deg ˆe
i
+ deg M
k
i, j = deg e
j
, 7.4
where M
k
i, j denotes the element at location i, j. We get
M
1
=
ab bc
cd ad
ac d
t t
c 1
t t
2
b t
t a
t t
2
t
3
,
7.5
for ∂
1
in our example. The reader may verify Equation 7.4 using this example for intuition, e.g., M
1
4, 4 = t
2
as deg ad − deg a = 2 − 0 = 2, according to
Table 7.1. Clearly, the standard bases for chain groups are homogeneous. We need to
represent ∂
k
:
C
k
→
C
k
−
1
relative to the standard basis for
C
k
and a homoge- neous basis for
Z
k
−
1
. We then reduce the matrix and read off the description of
H
k
according to our discussion in Section 7.3.1. We compute these represen- tations inductively in dimension. The base case is trivial. As
∂ ≡ 0,
Z
=
C
and the standard basis may be used for representing ∂
1
. Now, assume we have a matrix representation M
k
of ∂
k
relative to the standard basis {e
j
} for
C
k
and a homogeneous basis
{ ˆe
i
} for
Z
k
−
1
. For induction, we need to compute a ho- mogeneous basis for
Z
k
and represent ∂
k +1
relative to
C
k +1
and the computed basis. We begin by sorting basis ˆe
i
in reverse degree order, as already done in the matrix in Equation 7.5. We next transform M
k
into the column-echelon form ˜
M
k
, a lower staircase form shown in Figure 7.12 Uhlig, 2002. The steps have variable height, all landings have width equal to 1, and nonzero elements
may only occur beneath the staircase. A boxed value in the figure is a pivot and a row column with a pivot is called a pivot row column. From linear al-
gebra, we know that rank M
k
= rank
B
k
−
1
is the number of pivots in an echelon form. The basis elements corresponding to nonpivot columns form the desired
∗
∗ · · ·
∗ ∗
.. .
∗ ∗
· · · ∗
∗ · · ·
Fig. 7.12. The column-echelon form. An ∗ indicates a nonzero values and the pivots
are boxed.
basis for
Z
k
. In our example, we have
˜ M
1
=
cd bc
ab z
1
z
2
d t
c t
1 b
t t
a t
,
7.6
where z
1
= ad − cd − t · bc − t · ab and z
2
= ac − t
2
· bc − t
2
· ab form a homo- geneous basis for
Z
1
. The procedure that arrives at the echelon form is Gaussian elimination on the
columns, utilizing elementary column operations of types 1, 3 only. Starting with the left-most column, we eliminate nonzero entries occurring in pivot
rows in order of increasing row. To eliminate an entry, we use an elementary column operation of type 3 that maintains the homogeneity of the basis and
matrix elements. We continue until we either arrive at a zero column or we find a new pivot. If needed, we then perform a column exchange type 1 to
reorder the columns appropriately.
Theorem 7.4 echelon form The pivots in column-echelon form are the same as the diagonal elements in normal form. Moreover, the degree of the basis
elements on pivot rows is the same in both forms.
Proof Because of our sort, the degree of row basis elements ˆe
i
is monotonically decreasing from the top row down. Within each fixed column j, deg e
j
is a constant c. By Equation 7.4, deg M
k
i, j = c − deg ˆe
i
. Therefore, the degree of the elements in each column is monotonically increasing with row. We
may eliminate nonzero elements below pivots using row operations that do not change the pivot elements or the degrees of the row basis elements. We then
place the matrix in diagonal form with row and column swaps.
The theorem states that if we are only interested in the degree of the basis