mathematicalmarketing slide 1.1 linear algebra welcome to mar 6658 course title quantitative methods...
TRANSCRIPT
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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
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Slide 1.Slide 1.22Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
Ready to Get Going?
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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
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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
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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Slide 1.Slide 1.2525Linear AlgebraLinear Algebra
MathematicalMathematicalMarketingMarketing
The Inverse of a Product
Inverse of a Product: (AB)-1 = B-1 A-1
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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)