How does factor analysis work? eigenvalues and eigenvectors

factor.R

x <- data.frame(matrix(ncol = 2, byrow = T, c(
	-0.7,  0.2,
	2.1,  2.7,
	1.7,  2.3,
	1.4,  0.8,
	1.9,  2.0,
	1.8,  1.0,
	0.4, -0.4,
	1.1,  0.3,
	0.9,  0.4,
	-0.6,  0.7))
)
names(x) = c("x","y")
cov_x = cov(x)
eigen(cov_x)$vectors
# Eigenvalues suggest 1 factor
eigen(cov_x)$values
# Let's keep a feature matrix of just 1 factor: column 1
feature_vector = eigen(cov_x)$vectors[, 1, drop = F]
round(t(feature_vector) %*% t(x), 2)
# >       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
# > [1,] -0.35 3.39 2.83 1.56 2.76 1.98    0 0.99 0.92  0.07

Based on: this lecture