LINEAR MODELS AND MATRIX ALGEBRA

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LINEAR MODELS AND MATRIX ALGEBRA

• Chapter 4
• Alpha Chiang, Fundamental Methods of Mathematical
  Economics
• 3rd 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 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)

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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

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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

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Matrices as Arrays

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Example

• 6x1 3x2 x3 22
• x1 4x2-2x3 12
• 4x1 - x2 5x3 10

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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 elements of the matrix.
• Coefficient matrix
• Aaij

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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 a point 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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Example

• Qd Qs
• Qd a - bP
• Q s -c dP
• can be rewritten as
• 1Qd 1Qs 0
• 1Qd bP a
• 0 1Qs -dP -c

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In matrix form
Constant vector
Coefficient matrix
Variable vector
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Matrix Operations

• Addition and Subtraction matrices must have the
  same dimensions
• Example 1
• Example 2

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Matrix addition and subtraction

• In general
• Note that the sum matrix must have the same
  dimension as the component matrices.

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Matrix subtraction

• Subtraction
• Example

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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.

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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 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.

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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.

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Matrix Multiplication

• Exact Procedure

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Matrix multiplication

• Example 2x2, 2x2, 2x2

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Matrix multiplication

• Example 3x2, 2x1, 3x1

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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

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Matrix multiplication

• from 4.4, p56
• The product on the right is a column vector

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Matrix multiplication

• When we write Ax d, we have

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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

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Simple national income model

• Coefficient matrix, vector of variables, vector
  of constants
• To express it in terms of Axd,

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Simple national income model

• Thus, the matrix notation Axd would give
  us
• The equation Axd precisely represents the
  original equation system.

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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

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Digression on ? notation
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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

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Digression on ? notation

• Applying to each element of the product matrix
  CAB

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Digression on ? notation

• Extending to an m x n matrix, Aaik and an
• n x p matrix Bbkj, we may now write the
  elements of the m x p matrix ABCcij as
• or more generally,