LINEAR MODELS AND MATRIX ALGEBRA

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

2. 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)

3. 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

4. Matrices and Vectors Example 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

5. Matrices as Arrays

6. Example: • 6x1 + 3x2+ x3 = 22 x1 + 4x2+-2x3 =12 4x1 - x2 + 5x3 = 10

7. 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]

8. 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

9. 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.

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

11. Example Qd = Qs Qd = a - bP Q s= -c + dP • can be rewritten as 1Qd –1Qs = 0 1Qd + bP = a 0 +1Qs +-dP = -c

12. In matrix form… Constant vector Coefficient matrix Variable vector

13. Matrix Operations • Addition and Subtraction: matrices must have the same dimensions • Example 1: • Example 2:

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

15. Matrix subtraction • Subtraction • Example

16. 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.

17. 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.

18. 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.

19. Matrix Multiplication Exact Procedure

20. Matrix multiplication • Example : 2x2, 2x2, 2x2

21. Matrix multiplication • Example: 3x2, 2x1, 3x1

22. 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

23. Matrix multiplication • from 4.4, p56 • The product on the right is a column vector

24. Matrix multiplication • When we write Ax= d, we have

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

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

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

28. 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 index xj: summand

29. Digression on  notation:

30. 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. • general polynomial function

31. Digression on  notation: • Applying to each element of the product matrix C=AB

32. 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,