Download Chapter 1: Matrices

Chapter 1: Matrices
Definition 1: A matrix is a rectangular array of numbers arranged in horizontal
rows and vertical columns.
a11 a12

a21 a22
A  a31 a32

M
M


am1 am 2
a13
a23 K
a33 K
M
am 3 K
EXAMPLE:

K
4
0
5
 3
 1

3

7
6


 2  4
8  9
a1n 

a2n 
a3n 

M 
amn 

Matrix Operations: matrix
addition
Definition 2: Matrix Addition
Let A = ( ai j ) & B = ( bi j ) be mn matrices.
Then C = A + B is an mn matrix with elements ci j  ai j + bi j .
Example:
 1 3  0 0   1 3 
F  G   2 0    1 2   1 2 
 1 0  2 4   3 4 

 
 

Properties:
(A1) A + B = B + A
(A2) (A + B) + C = A + (B + C)
(A1) A + 0 = A
where 0 is a zero matrix (matrix consisting of only zero elements)
Matrix Operations:
scalar multiplication
Definition 3: Scalar Multiplication
Let A = ( ai j ) and λ  R.
Then C = λ A is an mn matrix with elements ci j  λ ai j .
Properties:
(S1) λ A = A λ
(S2) λ (A + B) = λ A + λ B
(S3) (λ1 + λ2) A = λ1 A + λ2 A
(S4) λ1 (λ2 A) =(λ1λ2) A
1.3 Matrix Multiplication
Definition: Matrix Multiplication
If A = ( ai j ) is an mn matrix and B = ( bi j ) is an np matrix,
Then the product AB is defined to be an mp matrix C = ( ci j ) where
ci j = ai 1 b1 j + ai 2 b2 j + … + ai n bn j
In other words,
ci j = ( Row i of A )  ( Column j of B )
Example:
1 1
 9 13 17 5 
4
6
8
2
 
 0 1


5
7
9
3






 1 0  5 7 9 3  4 6 8 2




Properties:
(M1) (AB)C = A(BC)
(M2) A(B + C) = AB + AC
(M3) (B + C)A = BA + CA
Matrix multiplication lacks commutativity.
Example 1:
 1 2   5 6   19 22 
 3 4   7 8    43 50 


 

 5 6   1 2   23 34 
 7 8   3 4    31 46 


 

Example 2:
 5 6  1 2 0   23 34 0 
 7 8  3 4 0    31 46 0 


 

1 2 0  5 6
 3 4 0   7 8  is not defined



Other properties that matrix multiplication lacks:
1) AB = 0 doesn’t imply A = 0 or B = 0
2) AB = AC doesn’t imply B = C
1.4 Special Matrices
Definition (matrix transpose): Given a matrix A, the transpose of A, denoted by
AT and read A-transpose, is obtained by changing all the rows of A into columns
of AT while preserving the order.
Examples:
T
T
6 2 
6 1
1 5    2 5 




6 2 
1 5   6 1 3
 2 5 8




3 8 
Properties of the transpose:
1) (AT)T = A
2) (λ A)T = λ AT
3) (A + B)T = AT + BT
4) (A B)T = BT AT
A symmetric matrix is a matrix that is equal to its transpose
while a skew symmetric matrix is a matrix that is equal to the negative of its transpose.
1.4 Special Matrices:
row-reduced form
Definition: A matrix is in row-reduced form if it satisfies four conditions:
(R1) All zero rows appear below nonzero rows when both types are present in
the matrix.
(R2) The first nonzero element in any nonzero row is unity.
(R3) All elements directly below (that is, in the same column but in succeeding
rows from) the first nonzero element of a nonzero row are zero.
(R4) The first nonzero element of any nonzero row appears in a later column
(further to the right) than the first nonzero element in any preceding row.
Example:
0
0

A  0

0
0
1 3  2 2
0 1 0 2
0 0 0 1

0 0 0 0
0 0 0 0
Row-reduced form: examples
• In row-reduced form
3  1
 1  5 2 1
0 1 0 5  
 0
0
0
1
3

2
0 0 1 3  
 

 0
0 0
1
4  0
0 0 0 0 
 
0
0
0
0
1

 0
• Not in row-reduced form
4
1 2  3
0 2

1

1


0 0
1  3
2
 1 2 1
0 0 0

0


0 1 2  4
0
1
0
0
0  1
0 2
1 3

0 0
1.4 Special Matrices:
identity matrice
Definition: A square matrix that has 1’s on the main diagonal and 0’s off the main
diagonal is called an identity matrix.
Example: A 3 × 3 identity matrix.
1 0 0
 0 1 0


0 0 1
Note: An identity matrix has the property that
AI = IA = A.
1.4 Special Matrices:
lower (upper) triangular matrices
Definition: A square matrix A=[aij] is called lower triangular
if aij=0 for j>i (that is, if all the elements above the main diagonal are zero)
and upper triangular
if aij=0 for j<i (that is, if all the elements below the main diagonal are zero).
Example:
2
0

A  0

0
0
1
3
2
3
1
0
0 1
0 0
4
0
0
0
0
2
2 
1

 3
1 
1.6 Vectors
Definition: A vector is a 1  n or n  1 matrix.
x2
v  ( x1 , x2 )
Magnitude:
If
|| v || 1 ,
v
|| v || x1  x2
2
v

x1
2
Is a UNIT vector
A nonzero vector is normalized if it is divided by its magnitude.
 x1
v
x2 



,
|| v ||  || v || || v || 
Is a unit vector
1.7 The geometry of vectors
Vector Addition
v  w  ( x1 , x2 )  ( y1 , y 2 )  ( x1  y1 , x2  y 2 )
v+w
v
w
Vector Subtraction
v  w  ( x1 , x 2 )  ( y1 , y 2 )  ( x1  y1 , x 2  y 2 )
v-w
v
w