Intermediate Mathematics for Economists ECON3272.
Computer Assignment 1. Matrix Algebra
In this course the open-source software R is used for numerical exercises. R includes a line editor but it is more convenient to use R with RStudio.
R is available in the computer labs and you may download it freely from the web onto your PC (http://www.r-project.org/).
For the installation of R and RStudio see:
Torfs, P. and Brauer, C. “A (very) short introduction to R”, 3 March 2014.
(http://cran.r-project.org/doc/contrib/Torfs+Brauer-Short-R-Intro.pdf)
RStudio splits the computer screen into four windows:
Editor window
• This window provides a text editor.
• Click Run or press CTRL+ENTER to send a line to the consol window where it is executed.
• You may edit and save the programs that you write in this window. Workspace/history window
• The workspace window shows the objects R has in its memory. You can edit their values by clicking on them.
• The history window provides a protocol of what has been typed.
Consol (command) window
• Commands are executed in this window.
• You may write directly into this window using the line editor. Plots/help/files/packages window
• R sends plots to this window.
• You may also use the help function, view files and install packages in this window.
A more detailed introduction to R can be found in:
Venables W.N. and Smith D.M. “An Introduction to R”, 16 April 2015.
(http://cran.r-project.org/doc/manuals/R-intro.pdf)
Data Input
Lines that start with # are comments that are ignored by R. Only copy the command lines without the comments. R is case sensitive; that is, x and X or gdp and Gdp indicate different objects.
# Start R.
# Input the vectors x and h.
x <- c(4, 3, 2, 7)
h <- c(5, -3, 1, 7, -6, 8, 1, 9, -6, -4, 0, 2)
# The assignment operator <-, which looks like an arrow, assigns
# the value to the right of it to the name on the left. The two
# vectors are now objects in the workspace of R.
# Rearrange the vector h into the 4×3 matrix A.
A <- array(h, dim=c(4,3))
# There are now three objects in the workspace. The following
# command shows the names of those objects.
objects()
# The output is
# [1] “A” “h” “x”
# Remove the vector h from the workspace.
rm(h)
objects()
# The output is
# [1] “A” “x”
# Show the two objects.
x; A
# The ouput is
# [1] 4 3 2 7
# [,1] [,2] [,3]
# [1,] 5 -6 -6
# [2,] -3 8 -4
# [3,] 1 1 0
# [4,] 7 9 2
# Note that matrix A is filled with numbers column by column. The
# elements of a vector are always listed in a row.
# Similarly, input the matrices B, C, D, E, F.
h <- c(3, 5, 5, -7, 4, -9, 8, 4, 0, 1, -7, 9)
B <- array(h, dim=c(3,4))
rm(h)
h <- c(1, 2, 4, -5, -7, 3, 8, 1, -6, 1, 1, 3)
C <- array(h, dim=c(3,4))
rm(h)
h <- c(7, -4, 2, 1, 9, -5, 0, 3, 6, 1, 4, 7, 0, 2, 1, -2)
D <- array(h, dim=c(4,4))
rm(h)
h <- c(3, 2, -1, 4, 5, -2, 7, -9, 6, 4, -2, 8, 0, 1, 9, 3)
E <- array(h, dim=c(4,4))
rm(h)
h <- c(-4, 6, -1, 3, 6, 2, 4, -8, -1, 4, 3, 0, 3, -8, 0, 7)
F <- array(h, dim=c(4,4))
rm(h)
# There are now six matrices and one vector in the workspace.
objects()
# Show all objects.
x; A; B; C; D; E; F
# Exercise 1
# Compute K=AB, L=BA, M=CD, N=DC, P=A(B+C), R=AB+AC. Display the
# results and provide comments where appropriate.
# For example, the R code for K and P is:
K <- A%*%B
K
P <- A%*%(B+C)
P
# Exercise 2
# Compute S=(AB)’-B’A’. The R function t produces the transpose of # a matrix. Comment.
S <- t(A%*%B)-t(B)%*%t(A)
S
# Exercise 3
# Find the determinants of matrices A, D and E. Call the
# determinants a, b, c. Comment where appropriate.
# Hint: One of these determinants does not exist. Also inspect the # columns of matrix E.
# Example:
a <- det(A)
a
# Exercise 4
# Compute the inverses of matrices A, D and E. Call them U, V, W. # Comment where appropriate.
# Hint: See the preceding question.
Example:
U <- solve(A)
U
# Exercise 5
# a) Compute the quadratic form x’Fx.
d <- t(x)%*%F%*%x
d
# The following code also works because R uses vectors in whatever
# way is multiplicatively coherent. Therefore, there is no need to
# transpose x; R does it automatically.
e <- x%*%F%*%x
e
# b) Compute the eigenvalues and eigenvectors of the symmetric
# matrix F.
ev <- eigen(F)
ev
# The eigenvalues are the row vector and the eigenvectors are the # columns of the matrix.
# c) Retrieve the matrix of eigenvectors.
Q <- ev$vec
# Compute J = Q’Q. Comment.
J = t(Q)%*%Q
J
# Hint: See the matrix whose columns are eigenvectors in the
# course handbook.
# d) What is the definiteness of the quadratic form x’Fx and
# matrix F?
# Exercise 6
objects()
# Remove all objects from the workspace.
rm(list=ls(all=TRUE))
# Check whether the workspace is now empty.
objects()
# a) Consider the linear equation system given in Bretscher
# (2009), Exercises 1.2, 17.
#
# Using matrix notations, the equation system is:
# Input the vector b and matrix A and display them. How to do this # is shown at the beginning of this assignment.
# b) Does the system have a unique solution?
# Hint: Compute the determinant of A. Is A invertible? For the R
# code see Exercise 3.
# c) Solve the system of equations.
# The following R code uses the inverse of A:
x <- solve(A)%*%b
# This code is, however, numerically inefficient and potentially
# unstable. A safer and more efficient code is:
solve(A,b)
# Note: In numerical linear algebra matrix inversion is avoided.
# There are other, more efficient ways to solve a linear equation
# system.
# Finally, do some housekeeping and remove all objects.
rm(list=ls(all=TRUE))
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