Fall 2003 Math 308/501–502
9 Matrix Methods for Linear Systems
9.3 Matrices and Vectors
Mon, 10/Nov
c
±
2003, Art Belmonte
Summary
(Read Chapter 11 of your lab manual for additional material.)
Types of matrices
An
m
×
n
matrix
is a rectangular collection of elements arranged
in
m
rows and
n
columns. These elements are real or complex
numbers or symbolic expressions. We’ll denote matrices with
uppercase boldface roman letters; e.g.,
A
,
B
, etc. Thus
A
=
±
a
ij
²
is a matrix whose
th entry is
a
.A
row vector
is a 1
×
n
matrix, whereas a
column vector
is an
n
×
1 matrix. Vectors will
be signiﬁed by lowercase boldface roman letters; e.g.,
v
,
x
,etc.A
matrix is square if
m
=
n
and diagonal if it is square with all
offdiagonal entries being zeros. The elements of a
zero matrix
or
a
zero vector
are all zeros. Either is denoted by
0
.
The Algebra of matrices
Matrix Addition and Scalar Multiplication
Provided that
matrices
A
and
B
have the same dimensions, their
sum
is given by
A
+
B
=
±
a
²
+
±
b
²
=
±
a
+
b
²
.
Scalar multiplication
is
deﬁned by
r
A
=
r
±
a
²
=
±
ra
²
scalar
is a real or complex
number or a symbolic expression. The expression

A
stands for
(

1
)
A
and matrix subtraction is deﬁned by
A

B
=
A
+
(

1
)
B
.
Here are some properties of matrix addition and scalar
multiplication.
A
+
(
B
+
C
)
=
(
A
+
B
)
+
C
A
+
B
=
B
+
A
A
+
0
=
A
A
+
(

A
)
=
0
r
(
A
+
B
)
=
r
A
+
r
B
(
r
+
s
)
A
=
r
A
+
s
A
r
(
s
A
)
=
(
rs
)
A
=
s
(
r
A
)
Matrix multiplication
Let
A
=
[
a
ik
]bean
m
×
n
matrix and
B
=
±
b
kj
²
an
n
×
p
matrix. The
matrix product
is deﬁned by
C
=
AB
=
A
*
B
=
±
c
²
,where
c
=
n
³
k
=
1
a
b
.Ino
ther
words,
c
is the dot product of the
i
th row of
A
with the
j
th
column of
B
. (NOTE: This is different that the
array product
in
MATLAB, deﬁned by
P
.
*
Q
=
±
p
²
.
*
±
q
²
=
±
p
q
²
,for
matrices
P
and
Q
having the same dimensions.) Here are some
properties of matrix multiplication. They are respectively known
as associativity, left distributivity, right distributivity, and another
instance of associativity.
(
AB
)
C
=
A
(
BC
)
A
(
B
+
C
)
=
AB
+
AC
(
A
+
B
)
C
=
+
BC
(
r
A
)
B
=
A
(
r
B
)
NOTE what is missing: commutativity! In general,
AB
6=
BA
,
even if both these products are deﬁned. (Indeed, one or both of
them may
not
be deﬁned.)
Matrices as Linear Operators
A consequence of the
aforementioned properties of matrix multiplication is that an
m
×
n
matrix
A
deﬁnes a
linear
operator from
R
n
to
R
m
(or else
C
n
to
C
m
):
A
(
x
+
y
)
=
Ax
+
Ay
and
A
(
r
x
)
=
r
Ax
.Fur
thermore
,
AB
constitutes a composition of the linear operators
A
and
B
and
is itself a linear operator; i.e.,
AB
=
A
◦
B
. Geometric examples
of linear operations that act on vectors are stectching, contracting,
rotation, and reﬂection.