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Welcome to MAR 6658
Mathematical
Marketing
Course Title
Quantitative Methods in Marketing IV:
Customer Metrics and Choice Modeling
Prerequisites
MAR 6507 or instructor permission
Instructor
Charles Hofacker
Meeting
M/W 10a-12p
Contact Info
Email: chofack @ cob.fsu.edu
Office: RBB 255
Hours: M/W 2.15p-3.15p
Grades
Two exams plus homework
Slide 1.1
Linear Algebra
Ready to Get Going?
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Slide 1.2
Linear Algebra
Vectors and Transposing Vectors
An m element column vector
a 1 
a 
a 2
 
 
a m 
Transpose the column
a  [a1 a 2  a m ].
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A q element row vector
b  [b1b2 bq ]
Transpose the row
b1 
b 
2
b   

 
b q 
Slide 1.3
Linear Algebra
A Matrix Is A Set of Vectors
•X is an n · m matrix
•First subscript indexes rows
•Second subscript indexes columns
 x11 x12
x
x 22
X   21
 

 x n1 x n 2
 x 1m 
 x 2m 

  

 x nm 
 {x ij }.
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Slide 1.4
Linear Algebra
The Transpose of a Matrix
 x11 x12
x
x 22
X   21
 

 x n1 x n 2
 x11 x 21
x
x 22
  12
 

 x 1n x 2 n
 x 1m 
 x 2m 

  

 x nm 
 x m1 
 xm2 

  

 x mn 

1 2 3 
A

 4 5 6
1 4 
A  2 5


3 6
14 2 
B

 3 14 
14 3 
B  

 2 14 
Note that (X')' = X
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Slide 1.5
Linear Algebra
The Dot Subscript Reduction Operator - Rows
We can display an intermediate amount of detail by
separately keeping track of each row:
x1  [ x11 x12  x1m ]
x2  [ x 21 x 22  x 2 m ]
 
xn  [ x n1 x n 2  x nm ]
So the matrix X becomes
x1 
x 
X   2 
 
 
xn 
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Slide 1.6
Linear Algebra
The Dot Subscript Reduction Operator – Columns
Or we can keep track of each column of X:
 x11 
 x12 
 x 1m 
x 
x 
x 
21
22




x1 
, x 2 
, , x  m   2 m 
 
 
 
 
 
 
 x n1 
x n 2 
 x nm 
So that X is
X  x1 x2  xm 
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Slide 1.7
Linear Algebra
The Equals Sign
A = B iff aij = bij for all i, j.
The matrices must have the same order.
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Slide 1.8
Linear Algebra
Some Special Matrices
d11
0
D


0
Diagonal
Scalar
Unit
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cI
1
0  0 
d 22  0 

   

0  d mm 
c
0



0
0
c

0




0
0



c 
1
1
1 
n m


1
1
1

1




1 
1 



1 
Slide 1.9
Linear Algebra
More Special Matrices
Null
Symmetric
Identity
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Marketing
0
0
0m  
n


0
d11
a



b
1
0
I


0
0
0

0




0
0



0
a  b 
d 22  c 

   

c  d mm 
0
1

0




0
0



1 
Slide 1.10
Linear Algebra
Matrix Addition
 Adding two matrices means adding corresponding
elements.
 The two matrices must be conformable.
CAB
{cij }  a ij  bij .
1 2 10 10 11 12
2 1  10 10  12 11

 
 

3 4 10 10 13 14
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Slide 1.11
Linear Algebra
Properties of Matrix Addition
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 Commutative:
A+B=B+A
 Associative:
A + (B + C) = (A + B) + C
 Identity:
A+0=A
Slide 1.12
Linear Algebra
Vector Multiplication
Vector multiplication works
with a row on the left
and a column on the right.
There are a lot of names for this:
•linear combination
•dot product
•scalar product
•inner product

 
      

 

ab  a 1 a 2
b1 
b 
 am   2 
 
 
b m 
 a 1b1  a 2 b 2    a m b m
m
  a i bi .
i 1
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Slide 1.13
Linear Algebra
Orthogonal Vectors
Two vectors x and y are said to be orthogonal if
xy  0
2
x =[2
1]
1
0
-2
-1
0
1
2
-1
-2
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Slide 1.14
Linear Algebra
Scalar Multiplication
1

3
10 
5

7

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2

4
10

30
 

6 50
8 70
20

40
60
80
Associative:
c1(c2A) = (c1c2)A
Distributive:
(c1 + c2) A = c1A + c2A
Slide 1.15
Linear Algebra
Matrix Multiplication
n
cij  aib j   a ik b kj
k
m
Cp m An n Bp
 5 4
2 3 
 1
C
0 2



 2  2 1 
 1 1
(1)5  (2)0  (3)  1 (1)4  (2)2  (3)1


 (2)5  (2)0  (1)  1 (2)4  (2)2  (1)1
 8 3


 9 5
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Slide 1.16
Linear Algebra
Partitioned Matrices
Visually, matrices act like scalars
AB  A1
 B1 
A 2     A 1B1  A 2 B 2
B 2 
And here is a little example
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a 12
a
  11
a
a
 21
22
 b11
a 13  
b 21
a 23  
b 31
a 
  11  b11
a 21 
b12
b12
b 22
b 32
a
b13    12
a 22
b13 
b 23 

b 33 
a 13  b 21
a 23  b 31
b 22
b 32
b 23 
b 33 
Slide 1.17
Linear Algebra
The Cross Product Matrix B
Keeping track of the columns of X
x1 
 x 
B  XX   2  x 1
 
 
xm 
x1 x 1 x1 x 2
 x x
x2 x 2
  2 1



xm x 1 xm x 2
x 2  x m 




x1 x m 
x2 x m 

 

xm x m 
 {x j x k }  {b jk }
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Slide 1.18
Linear Algebra
The Cross Product Matrix 2
Keeping track of the rows of X
B  XX  x 1
x 2  x n 
x1 
 x 
 2 
 
 
xn 
 x 1 x1  x 2 x2    x n xn 
n
  x i xi
i
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Slide 1.19
Linear Algebra
Properties of Multiplication
Scalar Multiplication:
 Commutative:
cA = Ac
 Associative:
A(cB) = (cA)B = c(AB)
Matrix Multiplication:
 Associative:
(AB)C = A(BC)
 Right Distributive:
A[B + C] = AB + AC
 Left Distributive:
[B + C]A = BA + CA
 Transpose of a Product
 Identity
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(BA)' = A'B'
IA = AI = A
Slide 1.20
Linear Algebra
The Trace of a Matrix
Tr S   s ii
i
Tr[AB] = Tr[BA] .
The theorem is applicable if both A and B
are square, or if A is m · n and B is n · m
Note that for a scalar s, Tr s = s.
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Slide 1.21
Linear Algebra
Solving a Linear System
Consider the following system in two unknowns:
a 11x 1  a 12 x 2  y1
a 21x 1  a 22 x 2  y 2
a 11 a 12   x 1   y1 
a
 x    y 
a
 21
 2
22   2 
Ax  y
The key to solving this is in the denominator below:
x1 
y1a 22  y 2 a 12
a 11a 22  a 12 a 21
|A|  a 11a 22  a 12a 21
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Slide 1.22
Linear Algebra
An Inverse for Matrices
ax = y
Ax = y
a-1ax = a-1y
A-1Ax = A-1y
1x = a-1y
Ix = A-1y
x = a-1y
x = A-1y
Scalars: One Equation and
One Unknown
Matrices: N Equations and
N Unkowns
We just need to find a matrix A-1 such that AA-1 = I.
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Slide 1.23
Linear Algebra
The Inverse of a 2 · 2
1
a 11 a 12 
1  a 22

a

|A|  a 21
 21 a 22 
 a 12 
a 11 
1
2 1
1  3  1

0 3
6  0 2


3

2 1  6
0 3  0


 6
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 1
6  1 0


2  0 1

6

Slide 1.24
Linear Algebra
The Inverse of a Product
Inverse of a Product:
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(AB)-1 = B-1 A-1
Slide 1.25
Linear Algebra
Quadratic Form
x' Ax  x 1
x2
 a 11
a
 x m   21


 a m1
a 12
a 22

a m2
 a 1m   x 1 
 a 2m   x 2 
 
    
 
 a mm   x m 
(Bilinear form is where the pre- and post-multiplying vectors are not necessarily identical)
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Slide 1.26
Linear Algebra