Environmental Engineering Reference
In-Depth Information
address single elements from the command window, or specify 2-dimensional
sub-arrays:
and edit those.
1.2.3 Basic Matrix Operations
It is expected that readers are already familiar with matrix operations and basics of
linear algebra. The purpose of the following is (1) to be a reminder for those, to
whom matrices are not (yet) part of daily practice and (2) to introduce the notation
used in the following chapters of this topic.
A matrix is a 2-dimensional array of numbers. A matrix has a certain number of
lines and columns, and the single numbers in the matrix are called elements
(sometimes the term 'entries' is used here as alternative). The matrix in (
1.1
) has
two lines and two columns, and the element in the second line and first column is 3.
A single number can be conceived as special case of a matrix with one line and one
column. Thus matrix algebra is a generalization of the usual calculations with single
numbers. However, in order to distinguish 'real' arrays from single numbers, bold
letters are used for matrices and vectors.
11
Basic operations as known from single numbers can be generalized for matrices.
Matrices can be added. The sum of the matrices A and B
0
@
1
A
0
@
1
A
a
11
a
12
:::
a
1
m
b
11
b
12
:::
b
1
m
a
21
a
22
:::
a
2
m
b
21
b
22
:::
b
2
m
A
¼
and B
¼
(1.2)
:::
:::
:::
:::
:::
:::
:::
:::
a
n
1
a
n
2
:::
a
nm
b
n
1
b
n
2
:::
b
nm
is given by:
0
1
a
11
þ b
11
a
12
þ b
12
:::
a
1
m
þ b
1
m
@
A
a
21
þ b
21
a
22
þ b
22
:::
a
2
m
þ b
2
m
A
þ
B
¼
(1.3)
:::
:::
:::
:::
a
n
1
þ b
n
1
a
n
2
þ b
n
2
:::
a
nm
þ b
nm
In order to add two matrices, both need to have the same number of lines and
columns. In each element of the matrix A + B, the sum of the corresponding
elements of A and B appears. One may also say that in order to obtain the element
11
A vector is a matrix consisting of one line or one column only. Terms as line-vectors or column-
vectors are used, too.
