#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) # Run this after restarting RStudio n <- 100 mu <- c(3,1) Sigma <- matrix(c(4,1,1,2),byrow=T,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=T,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 D1 <- as.matrix(read.table("Data1.txt",sep="\t",header=FALSE)) pairs(D1) center <- function(X){ pairs(X) ave <- apply(X,2,mean) cent <- sweep(X,2,ave,"-") return(cent) } C <- center(D1) apply(C,2,mean) cov(C) cor(C) eigen(cov(C))\$values eigen(cor(C))\$values eigen(cov(C))\$vectors eigen(cor(C))\$vectors