Introduction - Environmental Modeling - page 19

in the i- th row and j -th column of A + B, the elements in the i-th row and j-th
column of A and B have to be added:

ð A þ B Þ ij ¼ a ij þ b ij (1.4)

Example in MATLAB :

®

When the number of columns or the number of lines do not coincide,
MATLAB produces an error:

®

Clearly the subtraction of matrices is defined analogously. One may also for-
mally introduce subtraction by the definition that subtraction of B is the addition of
-B. As one may expect -B contains the negative of the elements of B and is the
inverse of B with respect to the addition operation. The generalizations of matrix
multiplication and division are slightly more complex.
It was already mentioned that there are several multiplication operations. Corre-
spondingly there are several division operations. Aside from scalar multiplication,
there are several matrix multiplications. The standard matrix multiplication for the
two matrices A and B, given by

0
@

1
A

0
@

1
A

a 11 a 12 ::: a 1 k b 11 b 12 ::: b 1 m

a 21 a 22 ::: a 2 k b 21 b 22 ::: b 2 m

A ¼ and B ¼ (1.5)

::: ::: ::: ::: ::: ::: ::: :::

a n 1 a n 2 ::: a nk b k 1 b k 2 ::: b km

in order to obtain a new matrix A B, is defined by the following formula:

X

k

ð A B Þ ij ¼ a ik b kj (1.6)

l¼ 1

This is a formula for the element in the i -th row and j -th column of the matrix AB.
Matrices can be multiplied if the first matrix has the same number of columns as the
second matrix has columns (inner dimension). In formula ( 1.6 ) that number is k .
Elements in lines of the first matrix are multiplied with columns of the second matrix,
and the products are summed in order to obtain an entry in the result matrix A

B.