Chapter 5 Matrices
From now on, we will typically write vectors as columns. Thus, when we write x ∈ F ^{n} we typically
mean
( )
x1
x = || .. ||
( . )
xn
We will also use the following convention.
( ) T
x1 ( x1 )
|| .. || ( ) ( )T | |
( . ) = x1 ⋅⋅⋅ xn , x1 ⋅⋅⋅ xn = ( ⋅⋅⋅ )
xn xn
The rules for adding and multiplying by a constant remain the same. To add, you add corresponding
entries and to multiply by a scalar, you multiply every entry by the scalar. Consider the following system of
equations:
x + y = 1
2x− y + z = 2
x + y = 1
Another way to write this is
( ) ( ) ( ) ( )
1 1 0 1
x |( 2 |) + y|( − 1|) +z |( 1 |) = |( 2 |)
1 1 0 1
That expression on the left is called a linear combination of the three vectors listed there whenever x,y,z
are numbers. Another way to write it is
( 1 1 0 ) ( x ) ( 1 )
| | | | | |
( 2 − 1 1 ) ( y ) = ( 2 )
1 1 0 z 1
The rows of the above matrix are
, , . The columns of this matrix
are
, , . It is called a 3
× 3 matrix because it has three rows and three columns.
More generally, we have the following definition.
Definition 5.0.1 An m × n matrix is a rectangular array of numbers which has m rows and n columns.
We write this as
( )
A11 A12 ⋅⋅⋅ A1n
|| A21 A22 ⋅⋅⋅ A2n ||
A = || .. .. .. ||
( . . . )
Am1 An2 ⋅⋅⋅ Amn
Thus the entry in the i ^{th} row and the j ^{th} column is denoted as A _{ij} . As suggested above,
( ) ( )
A11 A12 ⋅⋅⋅ A1n x1
|| A21 A22 ⋅⋅⋅ A2n || || x2 ||
Ax = || . . . || || . ||
( .. .. .. ) ( .. )
Am1 An2 ⋅⋅⋅ Amn xn
( ) ( ) ( )
A11 A12 A1n
|| A21 || || A22 || || A2n ||
= x1|| . || + x2|| . || + ⋅⋅⋅+ xn|| . ||
( .. ) ( .. ) ( .. )
Am1 An2 Amn
Note that A x is in F ^{m} and the i ^{th} entry of this vector A x is
∑n
Ai1x1 + Ai2x2 + ⋅⋅⋅+ Ainxn = Aijxj.
j=1
In other words, the i ^{th} entry of A x is the dot product of the i ^{th} row of A with the vector x .
Symbolically,
We like to write x to denote an n × 1 matrix which is often called a vector. Then x ^{T} will denote a 1 × n
matrix or row vector.
Example 5.0.2
( )( x1 )
1 2 1 | |
1 0 2 ( x2 )
x3
( ) ( ) ( ) ( )
= x1 1 +x2 2 + x3 1 = x1 + 2x2 + x3
1 0 2 x1 +2x3
Note that if A is m × n then A x is an m × 1 matrix provided x is n × 1. Thus A makes a vector in F ^{n}
into a vector in F ^{m} .
Example 5.0.3 Show the following:
( ) ( ) ( )
| 1 − 1 2 | | 1 | | 5 |
( 3 2 1 ) ( 2 ) = ( 10 )
2 3 − 3 3 − 1
Example 5.0.4 Write the system of equations
x +2y − z = 2
x − 3y + z = 1
in the form A x = b .
According to the above, this system can be written as
( ) ( x ) ( )
1 2 − 1 | | 2
1 − 3 1 ( y ) = 1
z
The following is the most fundamental observation about multiplying a matrix times a vector.
Theorem 5.0.5 Let A be an m × n matrix and let x , y be two vectors in F ^{n} with a,b two scalars.
Then
Proof: By the above definition and the way we add vectors,
∑ ∑ ∑
(A(ax +by ))i = Aij(axj + byj) = a Aijxj +b Aijxj = a(Ax )i + b(Ay )i = (aAx + bAy)i
j j j
Since the i ^{th} entries coincide, it follows that A
=
aA x +
bA y as claimed.
■
Definition 5.0.6 Define some special vectors e _{i} as follows:
th
◜-----1 in the i◞◟position---◝
e ≡ ( 0 ⋅⋅⋅ 0 1 0 ⋅⋅⋅ 0 )T
i
Thus in F ^{3} , we would have
( ) ( ) ( )
1 0 0
e1 = |( 0 |) ,e2 = |( 1 |) ,e3 = |( 0 |)
0 0 1
Observation 5.0.7 Let A be an m × n matrix. Then for e _{i} ∈ F ^{n} ,A e _{i} delivers the i ^{th} column of A . To see
this,
∑
(Aei) = Akj (ei) = Aki
k j j
because
_{j} = 0
unless j =
i when it is 1. Thus, for k arbitrary, the k ^{th} entry of A e _{i} is A _{ki} . Thus the
result of multiplying by e _{i} is
which is indeed the i ^{th} column. Another way to see this is to let
( )
0
|| .. ||
( ) ( )|| . ||
A = a1 ⋅⋅⋅ ai ⋅⋅⋅ an ,Aei = a1 ⋅⋅⋅ ai ⋅⋅⋅ an || 1 || = 1ai = ai
|( .. |)
.
0
the i ^{th} column of A .