LINEAR MODELS AND MATRIX ALGEBRA Chapter 4 Alpha

LINEAR MODELS AND MATRIX ALGEBRA Chapter 4 Alpha Chiang, Fundamental Methods of Mathematical Economics 3 rd edition

Why Matrix Algebra n n As more and more commodities are included in models, solution formulas become cumbersome. Matrix algebra enables to do us many things: n n n 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 n n 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 Vectors Example of a system of linear equations: c 1 P 1 + c 2 P 2 = -c 0 1 P 1 + 2 P 2 = - 0 In general, a 11 x 1 + a 12 x 2 +…+ a 1 n. Xn = d 1 a 21 x 1 + a 22 x 2 +…+ a 2 n. Xn = d 2 ……………… am 1 x 1 + am 2 x 2 +…+ amn. Xn = dm coefficients aij variables x 1, …, xn constants d 1, …, dm

Matrices as Arrays n

Example: n 6 x 1 + 3 x 2+ x 3 = 22 x 1 + 4 x 2+-2 x 3 =12 4 x 1 - x 2 + 5 x 3 = 10

Definition of Matrix n n 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]

Matrix Dimensions n n n 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

Vectors as Special Matrices n n one column : column vector one row: row vector n n usually distinguished from a column vector by the use of a primed symbol: Note that a vector is merely an ordered ntuple and as such it may be interpreted as a point in an n-dimensional space.

Matrix Notation n n Ax = d Questions: How do we multiply A and x? What is the meaning of equality?

Example n Qd = Q s Qd = a - b. P Q s= -c + d. P can be rewritten as 1 Qd – 1 Qs = 0 1 Qd + b. P = a 0 +1 Qs +-d. P = -c

In matrix form… Coefficient matrix Variable vector Constant vector

Matrix Operations n Addition and Subtraction: matrices must have the same dimensions n Example 1: n Example 2:

Matrix addition and subtraction n In general n Note that the sum matrix must have the same dimension as the component matrices.

Matrix subtraction n Subtraction n Example

Scalar Multiplication n n 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.

Matrix Multiplication n n 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.

Matrix Multiplication n n 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.

Matrix Multiplication Exact Procedure

Matrix multiplication n Example : 2 x 2, 2 x 2

Matrix multiplication n Example: 3 x 2, 2 x 1, 3 x 1

Matrix multiplication n n Example: 3 x 3, 3 x 3 Note, the last matrix is a square matrix with 1 s in its principal diagonal and 0 s everywhere else, is known as identity matrix

Matrix multiplication n from 4. 4, p 56 n The product on the right is a column vector

Matrix multiplication n When we write Ax= d, we have

Simple national income model n n Example : Simple national income model with two endogenous variables, Y and C Y = C + Io + Go C = a + b. Y can be rearranged into the standard format Y – C = Io – Go -b. Y + C = a

Simple national income model n Coefficient matrix, vector of variables, vector of constants n To express it in terms of Ax=d,

Simple national income model n n Thus, the matrix notation Ax=d would give us The equation Ax=d precisely represents the original equation system.

Digression on notation: n 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 index xj : summand

Digression on notation:

Digression on notation: n 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. -general polynomial function

Digression on notation: n Applying to each element of the product matrix C=AB

Digression on notation: n 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,