Notes Chapter 4 Part 1

8/4/2019 Notes Chapter 4 Part 1

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LINEAR MODELS ANDMATRIX ALGEBRA

Chapter 4Alpha Chiang, Fundamental Methods of

Mathematical Economics3rd edition


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Why Matrix Algebra As more and more commodities are included

in models, solution formulas become

cumbersome. Matrix algebra enables to do us many things:

provides a compact way of writing an equationsystem

leads to a way of testing the existence of asolution by evaluation of a determinant

gives a method of finding solution (if it exists)


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Catch Catch: matrix algebra is only applicable

to linear equation systems.

However, some transformation can bedone to obtain a linear relation.

y = axb

log y = log a + b log x


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Matrices and VectorsExample of a system of linear equations:

c1P1 + c2P2 = -c0

1P

1+

2P

2= -

0In general,

a11x1 + a12x2 ++ a1nXn = d1a21x1 + a22x2 ++ a2nXn = d2

am1x1 + am2x2 ++ amnXn = dmcoefficients aij

variables x1, ,xn

constants d1, ,d

m


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Matrices as Arrays

11 12 1 1 1

21 22 2 2 2

1 2

n

n

m m mn n

a a a x d

a a a x dA x d

a a a x dm


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Example: 6x1 + 3x2+ x3 = 22

x1 + 4x2+-2x3 =12

4x1 - x2 + 5x3 = 10

1

2

3

6 3 1 22

1 4 2 124 1 5 10

x

A x x dx


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Definition of Matrix A matrix is defined as a rectangular array of

numbers, parameters, or variables.

Members of the array are termed elementsof the matrix.

Coefficient matrix:

A=[aij]1, 2,...,

1, 2,...,

i m

j n


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Matrix Dimensions Dimension of a matrix = number of rows x

number of columns, m x n

m rows

n columns

Note: row number always precedes the

column number. this is in line with way the

two subscripts are in aij are ordered.

Special case: m = n, a square matrix


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Vectors as Special Matrices one column : column vector

one row: row vector

usually distinguished from a column vector by

the use of a primed symbol:

Note that a vector is merely an ordered n-

tuple and as such it may be interpreted as apoint in an n-dimensional space.


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Matrix Notation Ax = d

Questions: How do we multiply A and x?

What is the meaning of equality?


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ExampleQd = Qs

Qd = a - bP

Q s= -c + dP

can be rewritten as

1Qd1Qs = 01Qd + bP = a

0 +1Qs +-dP = -c


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1 1 0 0

1 0

0 1

d

s

Q

b Q a

d P c

In matrix form

Coefficient matrix Constant vectorVariablevector


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Matrix Operations Addition and Subtraction: matrices must

have the same dimensions

Example 1:

Example 2:

4 9 2 0 4 2 9 0 6 9

2 1 0 7 2 0 1 7 2 8

11 12 11 12 11 11 12 12

21 22 21 22 21 21 22 22

31 32 31 32 31 31 32 32

a a b b a b a b

a a b b a b a b

a a b b a b a b


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Matrix addition and

subtraction In general

Note that the sum matrix must have the same

dimension as the component matrices.

ij ij ij ij ij ija b c where c a b


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Matrix subtraction Subtraction

Example

19 3 6 8 19 6 3 8 13 5

2 0 1 3 2 1 0 3 1 3

ij ij ij ij ij ija b d where d a b


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Scalar Multiplication

To multiply a matrix by a number by a scalar is to multiply everyelement of that matrix by the given scalar.

Note that the rationale for the name scalar is that it scales up or down

the matrix by a certain multiple. It can also be a negative number.

3 1 21 7

7 0 5 0 35

1 1

11 1211 12 2 212 1 1

21 2221 22 2 2

a aa a

a aa a

11 12 1 11 12 1

21 22 2 21 22 2

1a a d a a d

a a d a a d


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Matrix Multiplication Given 2 matrices A and B, we want to find the product AB. The

conformability condition for multiplication is that the columndimension of A (the lead matrix) must be equal to the rowdimension of B ( the lag matrix).

BA is not defined since the conformability condition formultiplication is not satisfied.

11 12 1311 121 2 2 3 21 22 23x x

b b bA a a B

b b b


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Matrix Multiplication

In general, if A is of dimension m x n and B is ofdimension p x q, the matrix product AB will bedefined only ifn = p.

If defined the product matrix AB will have thedimension m x q, the same number of rows as thelead matrix A and the same number of columns asthe lag matrix B.

mxn pxq mxq

A B C


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Matrix Multiplication

Exact Procedure

11 12 13

11 121 2 2 3

21 22 23

11 12 131 3

11 11 11 12 21

12 11 12 12 22

13 11 13 12 23

where:

x x

x

b b bA a a B

b b b

AB c c c

c a b a bc a b a b

c a b a b


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Matrix multiplication

Example : 2x2, 2x2, 2x2

3 4 1 05 6 4 7

3( 1) 4(4) 3(0) 4(7) 13 285( 1) 6(4) 5(0) 6(7) 19 42

A and B

AB


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Example: 3x2, 2x1, 3x1

(3 2) (2 1) (3 1)

1 3 1(5) 3(9) 325

2 8 2(5) 8(9) 829

4 0 4(5) 0(9) 20x x x

A B AB


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Example: 3x3, 3x3, 3x3

Note, the last matrix is a square matrix with 1s in its principal diagonal

and 0s everywhere else, is known as identity matrix

3 1 2 0 1/ 5 3 /10

1 0 3 1 1/ 5 7 /10

4 0 2 0 2 / 5 1/10

3 1 4 9 7 20 1 0

5 10 1 0 01 0 6 3 0 3

0 0 0 0 1 05 10

0 0 14 0 4 12 0 2

0 0 05 10

A B

AB


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from 4.4, p56

The product on the right is a column vector

1

2

3

1 1 2 3

2 1 2 3

3 1 2 3

6 3 1 22

1 4 2 124 1 5 10

6 3 1 6 3

1 4 2 4 2

4 1 5 4 5

x

A x x dx

x x x x

Ax x x x x

x x x x


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When we write Ax= d, we have

1 2 3

1 2 3

1 2 3

6 3 22

4 2 12

4 5 10

x x x

x x x

x x x


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Simple national income model

Example : Simple national income model withtwo endogenous variables, Y and C

Y = C + Io + Go

C = a + bY

can be rearranged into the standard format

YC = IoGo

-bY + C = a


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Coefficient matrix, vector of variables, vector of constants

To express it in terms of Ax=d,

0 01 1

1

Y I G

A x db C a

1 1 1( ) ( 1)( )

1 1( )

Y Y C Y C Ax

b C bY C bY C


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Thus, the matrix notation Ax=d would give us

The equation Ax=d precisely represents the original equation

system.

0 0Y C I G

bY C a


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Digression on notation: Subcripted symbols helps in designating the locations

of parameters and variables but also lends itself to aflexible shorthand for denoting sums of terms, such

as those which arise during the process of matrixmultiplication.

j: summation index

xj: summand

3

1 2 3

1

j

j

x x x x


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Digression on notation:

7

3 4 5 6 7

3

0 1

0

i

i

n

k n

k

x x x x x x

x x x x


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Digression on notation: The application ofnotation can be readily extended to cases in which

the x term is prefixed with a coefficient or in which each term in the

sum is raised to some integer power.

3 3

1 2 3 1 2 3

1 1

3

1 1 2 2 2 3

1

0 1 20 1 2

0

2

0 1 2

( )j jj j

j j

j

n

j ni n

i

n

n

ax ax ax ax a x x x a x

a x a x a x a x

a x a x a x a x a x

a a x a x a x

-general polynomial function


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Digression on notation:Applying to each element of the product

matrix C=AB2

11 11 11 12 21 1 1

1

2

12 11 12 12 22 1 2

1

2

13 11 13 12 23 1 3

1

k k

k

k k

k

k k

k

c a b a b a b

c a b a b a b

c a b a b a b


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Digression on notation: Extending to an m x n matrix, A=[aik] and an

n x p matrix B=[bkj], we may now write the

elements of the m x p matrix AB=C=[cij] as

or more generally,

11 1 1 12 1 2

1 1

n n

k k k k

k k

c a b c a b

1,2,...,1 1,2,...,

1

n

i mij k kj j p

k

c a b

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