Graphics Reference
In-Depth Information
The matrix entry
a
ij
is located in the
i
-th row and
j
-th column of the array. An
m
×
n
matrix is said to be of
order m
n
,
A
is said to be a
square matrix
(of order
n
). A matrix of a single row is called a
row matrix
. Similarly, a matrix of a
single column is called a
column matrix
:
×
n
(“
m
by
n
”). If
m
=
⎡
⎤
b
1
b
2
.
b
m
⎣
⎦
=
a
1
a
n
,
A
a
2
···
B
=
.
A matrix is often viewed as consisting of a number of row or column matrices. Row
matrices and column matrices are also often referred to as
row vectors
and
column vec-
tors
, respectively. For a square matrix, entries for which
i
=
j
(that is,
a
11
,
a
22
,
...
,
a
nn
)
are called the
main diagonal
entries of the matrix. If
a
ij
=
0 for all
i
=
j
the matrix is
called
diagonal
:
⎡
⎣
⎤
⎦
a
11
0
···
0
0
a
22
···
0
A
=
.
.
.
.
.
.
.
00
···
a
nn
A square diagonal matrix with entries of 1 on the main diagonal and 0 for all other
entries is called an
identity matrix
, denoted by
I
. A square matrix
L
with all entries
above the main diagonal equal to zero is called a
lower triangular matrix
. If instead
all entries below the main diagonal of a matrix
U
are equal to zero, the matrix is an
upper triangular matrix
. For example:
⎡
⎤
⎡
⎤
⎡
⎤
100
010
001
200
1
154
021
003
⎣
⎦
,
L
⎣
⎦
,
U
⎣
⎦
.
=
=
−
20
=
I
−
50
−
1
The
transpose
of a matrix
A
, written
A
T
, is obtained by exchanging rows for columns,
and vice versa. That is, the transpose
B
of a matrix
A
is given by
b
ij
=
a
ji
:
⎡
⎤
5
.
52
−
30
⎣
⎦
,
B
A
T
A
=
−
3
−
1
=
=
2
−
1
−
4
0
−
4
