## 2.23
A <- matrix(c(4,-5,-1,7,-2,3,-1,4,-3,8,2,6), 4,3, byrow=TRUE)
A.Aprime <- A%*%t(A)
Aprime.A <- t(A) %*% A
U <- eigen(A.Aprime)$vectors[,1:3]
V <- eigen(Aprime.A)$vectors
D <- diag(sqrt(eigen(Aprime.A)$values))
U %*% D %*% t(V)
D.signed <- zapsmall( t(U) %*% A %*% V)
U %*% D.signed %*% t(V)
V <- V %*% sign(D.signed)
U %*% D %*% t(V)