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Welcome to MAR 6658. Course Title Quantitative Methods in Marketing IV: Psychometric and Econometric Techniques Prerequisites MAR 6507 or instructor permission Instructor Charles Hofacker Meeting Tue 1:00-5:00 Contact Info Email : chofack @ cob.fsu.edu Office : RBB 255 - PowerPoint PPT Presentation
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Slide 1.Slide 1.11Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
Welcome to MAR 6658
Course Title Quantitative Methods in Marketing IV:Psychometric and Econometric Techniques
Prerequisites MAR 6507 or instructor permission
Instructor Charles Hofacker
Meeting Tue 1:00-5:00
Contact Info Email: chofack @ cob.fsu.eduOffice: RBB 255Hours: T/ R 11:00-12:00
Grades Two exams plus homework
Slide 1.Slide 1.22Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
Ready to Get Going?
Slide 1.Slide 1.33Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
Vectors and Transposing Vectors
m
2
1
a
a
a
a
]bbb[q21
b
An m element column vector A q element row vector
Transpose the column Transpose the row
].aaa[m21
a
q
2
1
b
b
b
b
Slide 1.Slide 1.44Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
A Matrix Is A Set of Vectors
.}x{
xxx
xxx
xxx
ij
nm2n1n
m22221
m11211
X
•X is an n · m matrix•First subscript indexes rows•Second subscript indexes columns
Slide 1.Slide 1.55Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
The Transpose of a Matrix
mnn2n1
2m2212
1m2111
nm2n1n
m22221
m11211
xxx
xxx
xxx
xxx
xxx
xxx
X
63
52
41
654
321
A
A
142
314
143
214
B
B
Note that (X')' = X
Slide 1.Slide 1.66Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
The Dot Subscript Reduction Operator - Rows
]xxx[
]xxx[
]xxx[
nm2n1nn
m222212
m112111
x
x
x
We can display an intermediate amount of detail by separately keeping track of each row:
So the matrix X becomes
n
2
1
x
x
x
X
Slide 1.Slide 1.77Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
The Dot Subscript Reduction Operator – Columns
Or we can keep track of each column of X:
nm
m2
m1
m
2n
22
12
2
1n
21
11
1
x
x
x
,,
x
x
x
,
x
x
x
xxx
So that X is
m21 xxxX
Slide 1.Slide 1.88Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
The Equals Sign
A = B iff aij = bij for all i, j.
The matrices must have the same order.
Slide 1.Slide 1.99Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
Some Special Matrices
Diagonal
Scalar cI
Unit 1
111
111
111
mn
1
mm
22
11
d00
0d0
00d
D
c00
0c0
00c
Slide 1.Slide 1.1010Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
More Special Matrices
Null
Symmetric
Identity
mm
22
11
dcb
cda
bad
100
010
001
I
000
000
000
mn
0
Slide 1.Slide 1.1111Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
Matrix Addition
.ba}c{ijijij
BAC
Adding two matrices means adding correspondingelements.
The two matrices must be conformable.
1413
1112
1211
1010
1010
1010
43
12
21
Slide 1.Slide 1.1212Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
Properties of Matrix Addition
Commutative: A + B = B + A
Associative: A + (B + C) = (A + B) + C
Identity: A + 0 = A
Slide 1.Slide 1.1313Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
Vector Multiplication
.ba
bababa
b
b
b
aaa
m
1iii
mm2211
m
2
1
m21
ba
Vector multiplication works with a row on the leftand a column on the right.
There are a lot of names for this:
•linear combination•dot product•scalar product•inner product
Slide 1.Slide 1.1414Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
Orthogonal Vectors
-2
-1
0
1
2
-2 -1 0 1 2
x =[2 1]
0yx
Two vectors x and y are said to be orthogonal if
Slide 1.Slide 1.1515Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
Scalar Multiplication
8070
6050
4030
2010
87
65
43
21
10
Associative: c1(c2A) = (c1c2)A
Distributive: (c1 + c2) A = c1A + c2A
Slide 1.Slide 1.1616Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
Matrix Multiplication
n
kkjikjiij bac ba
pnnmpm BAC
59
38
1)1(2)2(4)2(1)1(0)2(5)2(
1)3(2)2(4)1(1)3(0)2(5)1(
11
20
45
122
321C
Slide 1.Slide 1.1717Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
Partitioned Matrices
2211
2
1
21 BABAB
BAAAB
333231
232221
2322
1312
131211
21
11
333231
232221
131211
232221
131211
bbb
bbb
aa
aabbb
a
a
bbb
bbb
bbb
aaa
aaa
Visually, matrices act like scalars
And here is a little example
Slide 1.Slide 1.1818Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
The Cross Product Matrix B
}b{}{ jkkj
xx
xxxxxx
xxxxxx
xxxxxx
xxx
x
x
x
XXB
mm2m1m
m22212
m12111
m21
m
2
1
Keeping track of the columns of X
Slide 1.Slide 1.1919Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
The Cross Product Matrix 2
n
iii
2211
n
2
1
21
xx
xxxxxx
x
x
x
xxxXXB
nn
n
Keeping track of the rows of X
Slide 1.Slide 1.2020Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
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 (BA)' = A'B'
Identity IA = AI = A
Slide 1.Slide 1.2121Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
The Trace of a Matrix
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.
i
iisTr S
Slide 1.Slide 1.2222Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
Solving a Linear System
yAx
2
1
2
1
2221
1211
2222121
1212111
y
y
x
x
aa
aa
yxaxa
yxaxa
21122211
1222211 aaaa
ayayx
Consider the following system in two unknowns:
The key to solving this is in the denominator below:
21122211 aaaa|| A
Slide 1.Slide 1.2323Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
An Inverse for Matrices
ax = y
a-1ax = a-1y
1x = a-1y
x = a-1y
Ax = y
A-1Ax = A-1y
Ix = A-1y
x = A-1y
Scalars: One Equation andOne Unknown
Matrices: N Equations andN Unkowns
We just need to find a matrix A-1 such that AA-1 = I.
Slide 1.Slide 1.2424Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
The Inverse of a 2 · 2
20
13
6
1
30
121
10
01
6
2
6
0
6
1
6
3
30
12
1121
1222
1
2221
1211
aa
aa
|A|
1aa
aa
Slide 1.Slide 1.2525Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
The Inverse of a Product
Inverse of a Product: (AB)-1 = B-1 A-1
Slide 1.Slide 1.2626Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
Quadratic Form
m
2
1
mm2m1m
m22221
m11211
m21
x
x
x
aaa
aaa
aaa
xxx
Axx'
(Bilinear form is where the pre- and post-multiplying vectors are not necessarily identical)
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