Welcome to MAR 6658

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



Ready to Get Going?



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



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



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



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



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



The Equals Sign

A = B iff aij = bij for all i, j.

The matrices must have the same order.



Some Special Matrices

Diagonal

Scalar cI

Unit 1

111

111

111

mn

1

mm

22

11

d00

0d0

00d

D

c00

0c0

00c



More Special Matrices

Null

Symmetric

Identity

mm

22

11

dcb

cda

bad

100

010

001

I

000

000

000

mn

0



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



Properties of Matrix Addition

Commutative: A + B = B + A

Associative: A + (B + C) = (A + B) + C

Identity: A + 0 = A



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



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



Scalar Multiplication

8070

6050

4030

2010

87

65

43

21

10

Associative: c1(c2A) = (c1c2)A

Distributive: (c1 + c2) A = c1A + c2A



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



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



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



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



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



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



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



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.



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



The Inverse of a Product

Inverse of a Product: (AB)-1 = B-1 A-1



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)

Welcome to MAR 6658

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