MATRIX

library(mvtnorm)
library(matlib)
library(ggplot2)

sigma <- matrix(c(4,2,2,3), ncol = 2, nrow = 2)
mu <- c(1, 2)

n <- 1000

set.seed(123)

x <- rmvnorm(n = n, mean = mu, sigma = sigma)

d <- data.frame(x)

p2 <- ggplot(d, aes(x = X1, y = X2)) +
geom_point(alpha = .5) +
geom_density_2d()
p2

y <- x - mu

E <- eigen(sigma)
E$vectors

##            [,1]       [,2]
## [1,] -0.7882054  0.6154122
## [2,] -0.6154122 -0.7882054

y <- y %*% t(inv(E$vectors))

dd <- data.frame(y)

p3 <- ggplot(dd, aes(x = X1, y = X2)) +
geom_point(alpha = .5) +
geom_density_2d()
p3

#own number

sigma <- matrix(c(8,4,4,6), ncol = 2, nrow = 2)
mu <- c(4, 4)

n <- 500

set.seed(465)

x <- rmvnorm(n = n, mean = mu, sigma = sigma)

d <- data.frame(x)

p2 <- ggplot(d, aes(x = X1, y = X2)) +
geom_point(alpha = .3) +
geom_density_2d()
p2

y <- x - mu

E <- eigen(sigma)
E$vectors

##            [,1]       [,2]
## [1,] -0.7882054  0.6154122
## [2,] -0.6154122 -0.7882054

y <- y %*% t(inv(E$vectors))

dd <- data.frame(y)

p3 <- ggplot(dd, aes(x = X1, y = X2)) +
geom_point(alpha = .3) +
geom_density_2d()
p3