##############################

# Define matrix as column (document) array

A = matrix ( c(
    c (1, 2, 3, 4, 5),  # First document
    c (3, 2, 1, 4, 3),
    c (4, 2, 3, 1, 2),
    c (2, 2, 4, 3, 5),
    c (4, 4, 3, 2, 1),
    c (3, 4, 3, 4, 4)), # Sixth document
    nrow = 5, # terms
    ncol = 6) # documents

# Print matrix A
A

# Singular value decomposition
SVD = svd (A)

# Sigmas are positive and non-decreasing
S = diag(SVD$d)	      # Sigma
S1 = diag((SVD$d)^-1) # Ita inverse
U = SVD$u
V = SVD$v

# Print the matrices
S
U
V

# Check the column orthonormality of U and V
t(U) %*% U
t(V) %*% V

# Check that A is reproduced
U %*% S %*% t(V)

##########################################
#
# Handle a query
#

# Define a query as a document
q = c (3, 2, 3, 5, 4)

# Compute and print direct document similarity to q
Sim = t(A) %*% q
Sim

# Convert query q into SVD frame
q1 = S1 %*% t(U) %*% q

# Print new coordinates
q1

# check that similarity is the same as above
V %*% S^2 %*% q1

#######################################
#
# LSI
#

# Reduce rank to k terms
k = 3

# Cut matrices off
Uk = U[,1:k]
Vk = V[,1:k]
Sk = diag(SVD$d[1:k])
Sk1 = diag((SVD$d[1:k])^-1)

# Print new matrices
Uk
Vk
Sk
Sk1

# Print approximate document-term matrix
# to see how it changed
Ak = Uk %*% Sk %*% t(Vk)

# Project query onto restricted LSI space
qk = Sk1 %*% t(Uk) %*% q

# Print it
qk

# Compute approximate similarity
Simk = Vk %*% Sk^2 %*% qk

# Print all relevant results
A
Ak
Sim
Simk
q
qk

############################################