# svd1.r                 September 2007, 2009
 #
 # some examples of use of svd to do Singular Value Decomposition
 #
 # first example -- singular design matrix
  X <- matrix(1,6,4)
  X[,2] <- 1:6
  X[,3] <- 19:24    # dependent on first two
  X[,4] <- (1:6)*(1:6)
  X                 # should look familiar
 #                  first qr
  qr(X)             # what does QR do in this case?
 #                  next svd
  svd(X)
 #                  inner product matrix
  xpx <- t(X) %*% X
  print("X'X")
  xpx
  eigen(xpx,symmetric=TRUE)            # eigenvectors and values of X'X
  xxp <- X %*% t(X)
  print("XX'")
  xxp 
  exxp <- eigen(xxp,symmetric=TRUE)    # vector and values of XX'
  exxp$values                          # eigenvalues are same except 0's
  sqrt(exxp$values)                    # singular values
  exxp$vectors                         # other singular vectors
 # done
  rm(list=ls())
 q()