# Create a simple 3 X 2 matrix X and find its principal components and
# singular values using the R function svd()

#   Corrected (see below)
#   Corrected again (extract named elements from the "results" list)

#    For the class
#  https://www.math.nyu.edu/faculty/goodman/teaching/MathFin2019/MathFinance.html
#    For use with Assignment 8, posted there
#

# The matrix X is
#      1 2
#      0 1
#      1 0

X = matrix( nrow = 3, ncol = 2)
# X[1,1] = 1
# X[1,2] = 2
# X[2,1] = 0
# X[2,2] = 1
# X[3,1] = 1
# X[3,2] = 0

#    Credit to Leo Lin for finding a mistake in the statements commented
#    out above.

X[1,1] = 1
X[1,2] = 2
X[2,1] = 0
X[2,2] = 2
X[3,1] = 1
X[3,2] = 0

result = svd(X)      # svd() returns a named list with the elements of PCA

#    R has a "data structure" called "named list".
#    You get the element named "name" from a list object named "list"
#    with: list$name.

sig    = result$d    # the singular values are named "d"
U      = result$u    # the left principal components are the columns of "u"
v      = result$v    # the right principal components are the columns of "v"