LINEAR MODELS AND MATRIX ALGEBRA

Chapter 4Alpha Chiang, Fundamental Methods of Mathematical

Economics3rd edition

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

equation system leads to a way of testing the existence of a

solution by evaluation of a determinant gives a method of finding solution (if it

exists)

Catch Catch: matrix algebra is only

applicable to linear equation systems.

However, some transformation can be done to obtain a linear relation.

y = axb

log y = log a + b log x

Matrices and VectorsExample of a system of linear equations:

c1P1 + c2P2 = -c0

1P1 + 2P2 = -0

In general, a11 x1 + a12 x2 +…+ a1nXn = d1

a21 x1 + a22 x2 +…+ a2nXn = d2

………………………………am1 x1 + am2 x2 +…+ amnXn = dm

coefficients aij

variables x1, …,xn constants d1, …,dm

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 da a a x d

A x d

a a a x dm

Example: 6x1 + 3x2+ x3 = 22

x1 + 4x2+-2x3 =124x1 - x2 + 5x3 = 10

1

2

3

6 3 1 221 4 2 124 1 5 10

xA x x d

x

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

numbers, parameters, or variables. Members of the array are termed elements of the matrix.

Coefficient matrix: A=[aij] 1,2,...,

1, 2,...,i mj n

Matrix Dimensions Dimension of a matrix = number of rows x

number of columns, m x nm rowsn 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

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 a point in an n-dimensional space.

Matrix Notation Ax = d

Questions: How do we multiply A and x? What is the meaning of equality?

ExampleQd = Qs

Qd = a - bPQ s= -c + dP

can be rewritten as1Qd –1Qs = 01Qd + bP = a0 +1Qs +-dP = -c

1 1 0 01 00 1

d

s

Qb Q ad P c

In matrix form…

Coefficient matrix Constant vectorVariable vector

Matrix Operations Addition and Subtraction: matrices must

have the same dimensions Example 1:

Example 2:

4 9 2 0 4 2 9 0 6 92 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 ba a b b a b a ba a b b a b a b

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

Matrix subtraction Subtraction

Example19 3 6 8 19 6 3 8 13 52 0 1 3 2 1 0 3 1 3

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

Scalar Multiplication To multiply a matrix by a number – by a scalar – is to multiply

every element 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 77

0 5 0 35

1 111 1211 12 2 21

2 1 121 2221 22 2 2

a aa aa aa a

11 12 1 11 12 1

21 22 2 21 22 2

1a a d a a da a d a a d

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

AB. The conformability condition for multiplication is that the column dimension of A (the lead matrix) must be equal to the row dimension of B ( the lag matrix).

BA is not defined since the conformability condition for multiplication is not satisfied.

11 12 13

11 121 2 2 3 21 22 23x x

b b bA a a B

b b b

Matrix Multiplication In general, if A is of dimension m x n and B is

of dimension p x q, the matrix product AB will be defined only if n = p.

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

mxn pxq mxq

A B C

Matrix MultiplicationExact Procedure

11 12 1311 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 bc a b a b

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

Matrix multiplication 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 xA B AB

Matrix multiplication 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 3/101 0 3 1 1/ 5 7 /104 0 2 0 2 / 5 1/10

3 1 4 9 7 20 1 05 10 1 0 0

1 0 6 3 0 30 0 0 0 1 05 10

0 0 14 0 4 12 0 20 0 05 10

A B

AB

Matrix multiplication 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 1 221 4 2 124 1 5 10

6 3 1 6 31 4 2 4 24 1 5 4 5

xA x x d

x

x x x xAx x x x x

x x x x

Matrix multiplication When we write Ax= d, we have

1 2 3

1 2 3

1 2 3

6 3 224 2 12

4 5 10

x x xx x x

x x x

Simple national income model Example : Simple national income model with

two endogenous variables, Y and C

Y = C + Io + GoC = a + bY

can be rearranged into the standard format

Y – C = Io – Go -bY + C = a

Simple national income model Coefficient matrix, vector of variables, vector of

constants

To express it in terms of Ax=d,

0 01 11

Y I GA x d

b C a

1 1 1( ) ( 1)( )1 1( )

Y Y C Y CAx

b C bY C bY C

Simple national income model Thus, the matrix notation Ax=d would give us

The equation Ax=d precisely represents the original equation system.

0 0Y C I GbY C a

Digression on notation: Subcripted symbols helps in designating the

locations of parameters and variables but also lends itself to a flexible shorthand for denoting sums of terms, such as those which arise during the process of matrix multiplication.

j: summation indexxj: summand

3

1 2 31

jj

x x x x

Digression on notation:

7

3 4 5 6 73

0 10

ii

n

k nk

x x x x x x

x x x x

Digression on notation: The application of notation 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 31 1

3

1 1 2 2 2 31

0 1 20 1 2

0

20 1 2

( )j jj j

j jj

nj n

i ni

nn

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

Digression on notation: Applying to each element of the

product matrix C=AB2

11 11 11 12 21 1 11

2

12 11 12 12 22 1 21

2

13 11 13 12 23 1 31

k kk

k kk

k kk

c a b a b a b

c a b a b a b

c a b a b a b

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 21 1

n n

k k k kk k

c a b c a b

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

1

ni m

ij k kj j pk

c a b