Page 138 - Becoming Metric Wise

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Publication and Citation Analysis

an author-article matrix we will derive the number of authors, the number
of contributions of a particular author and the number of common contri-
butors. This section is based on (Krauze & McGinnis, 1979).
Assume that an (m,n) matrix C, with elements c ij , is a citation matrix.
This means that c ij 5 1 if article i cites document j and that c ij 5 0 if this is
not the case. Recall that c ij denotes the element, entry or cell (three terms
used here as synonyms) situated at the intersection of the i-th row and
j-th column.
We know from the dimensions of the citation matrix that we consider a
collection of m articles, citing in total n different documents. Mathematical
relations become easy if one knows how to multiply matrices. Readers
who know this or are not interested may safely skip the next paragraph.

Matrices and how to Multiply Them
Let A and B be matrices. Let A be an (m,n)-matrix and let B be a (k,l)-
matrix. The dimensions of a matrix can be any couple of strict positive
natural numbers. An (m,1) is a column matrix (or column vector); an
(1,l)-matrix is a row matrix (or row vector); an (1,1)-matrix is just a
number. If one or both of the dimensions of a matrix P is known to be
one, then the corresponding 1 is not written: one writes P j,1 simply as P j
(or p j , using the convention to write a matrix with a majuscule and its
entries with the corresponding minuscule). For further use we note
t
that if C is a matrix then C is its transpose. This means that rows and
t
columns have been interchanged. If C is an (m,n)-matrix then C is an
t
(n,m)-matrix and (C) ij 5 c ij 5 (C ) ji .
First we note that the matrix multiplication A B is in general not
the same as B A. It might even occur that one of these two multiplica-
tions is possible (is defined), while the other one is not. Indeed A B is
defined only if n 5 k; similarlyB A is only defined if l 5 m. Assuming
that A B is defined, hence that A is an (m,n) matrix and B is an (n,l)-
matrix then the resulting matrix C 5 A B has dimensions (m,l). The
n
element c ij of the matrix C is then defined as: c ij 5 P a ik b kj . This is all
k51
one has to know to understand matrix multiplication.

1 0 0 1 21
An example: If A 5 and B 5 then
212 4 6 10

0 1 21
A B 5 C 5 .
8 11 21