# a)

# setwd("...")
help(c)
help(matrix)


# b)
a <- matrix(c(2, 1, 5, -2, 7, 0, 5, -8, -1), ncol = 3, byrow = TRUE)

x1 <- c(8, -4, 2)
x2 <- matrix(c(8, -4, 2), nrow = 1)
b <- c(3, 10, -19)

y1 <- x1 %*% solve(a) + b
y2 <- x2 %*% solve(a) + b

y3 <- solve(a) %*% x1  + b # Note that R does not give an error here
y4 <- solve(a) %*% x2  + b # Here, R gives an error

# Note that when multiplying vectors and matrices, the safe way
# is to always use variables of the class matrix

# c)
set.seed(123)

# install.packages("mvtnorm") # Run this only once

library(mvtnorm)

n <- 100
mu <- c(3, 1)
sigma <- matrix(c(4, 1, 1, 2), byrow = TRUE, ncol = 2)

x <- rmvnorm(n, mu, sigma)
plot(x)

# d)
mx <- apply(x, 2, mean)
colMeans(x)
sx <- cov(x)

sum(diag(sx)) - sum(eigen(sx)$values)
prod(eigen(sx)$values) - det(sx)

# e)
b <- c(3, 1)
a <- matrix(c(1, 2, 3, 1), byrow = TRUE, ncol = 2)
y <- sweep(x %*% t(a), 2, b, "+")

# another way
ones <- rep(1, n)
y2 <- x %*% t(a) + ones %*% t(b)

# When comparing if multivariate expressions are the same,
# use, e.g., the Frobenius norm
norm(colMeans(y) - a %*% colMeans(x) - b, type = "F")
norm(cov(y) - a %*% cov(x) %*% t(a), type = "F")

# f)

# Here, we make a conversion to type matrix,
# note that many of the basic matrix operations are not
# available for variables of type data.frame
d <- as.matrix(read.table("data/data.txt", sep = "\t", header = FALSE))
pairs(d)

center <- function(x) {
  ave <- apply(x, 2, mean)
  sweep(x, 2, ave, "-")
}

cent <- center(d)
apply(cent, 2, mean)

cov(cent)
cor(cent)

eigen(cov(cent))$values
eigen(cor(cent))$values

eigen(cov(cent))$vectors
eigen(cor(cent))$vectors