Page 139 - Becoming Metric Wise

130 Becoming Metric-Wise

0 1
1
Let D 5 @ A and suppose that we want to calculate D D. First we
2
t
0
note that this multiplication is mathematically possible as the number of
t
columns of D is equal to the number of rows of D. The result of this
0 1
1

2
matrix multiplication is: 1 2 0 @ A 5 1 1 4 1 0 5 5 (this is a
0
(1,1)-matrix or a number, sometimes referred to as a scalar when matri-
ces, vectors and numbers (scalars) are used in the same context).
t

Note that also D D exists. This is a (3,3)-matrix obtained as follows:
0 1 0 1
1 1 2 0

2
@ A 1 2 0 5 @ 2 4 0 A
0 0 0 0
Using matrix multiplication we now prove four propositions showing
how to derive the number of references of a given article, the number of
articles written by a given author, the number of citations received by a
given paper and the number of coauthors of a given paper, when a larger
article-article matrix C or an author-article matrix W is given.

Proposition 1: Given the citation matrix C, the number of references of
P n t
a given article d i is j51 ij 5 C Uð Þ 5 C C Þ , where U is the col-
c
ð
ii
i
umn vector completely consisting of 1’s.
Proof: This result is easy to see since the C matrix consists of zeros
and ones, ones if the corresponding cell is occupied and zero otherwise.
P n
c
Keeping the row i fixed j51 ij is just the number of ones in the i-th
row. This is the number of times document d i has a reference, or the total
number of references of document d i .
Now C U is an (m,1) matrix, i.e., a column vector. C UÞ is the
ð
i
i-th element of this column vector. It is equal to:
P n P n P n
c
c
c
ð C UÞ 5 j51 ij U j 5 j51 ij 1 5 j51 ij .
i
t
Similarly C C Þ 5 P n j51 ðÞ ij ð t ji P n j51 ij c ij 5 P n j51 ij . The last
C : C Þ 5
c
c
ð
ii
2
2
equality follows from the facts that 1 5 1 and 0 5 0.
Next we consider an author-article matrix W.
Assume that the (m,n)-matrix W, with elements w ij , is an author-
article matrix. This means that w ij 5 1 if author i has authored (as sole
author or as coauthor) article j, and zero if this is not the case. The