I have a question that is not directly related to R ... but I try to do it in R ;-) :
I would like to generate a matrix Q satisfying (for a given Z, X and W) the two following conditions:
t(Q)%*%Q=Z (1) XQ=W (2)
where: Q is m rows and r columns X is p rows and m columns D is p rows and r columns C is r rows and r columns with m>p,r
e.g: m=6, p=2 r=3
Z=matrix(c(1,.2,.5,.2,1,.45,.5,.45,1),3,3) X=matrix(c(.1,.3,.5,.6,.2,.1,.8,1,.4,.2,.2,.9),2,6) W=matrix(c(0,.8,.4,.6,.2,0),2,3)
#Create a matrix satisfying (1) is easy:
A=matrix(runif(18),6,3) Q1=svd(A)$u%*%chol(Z)
#For the second condition (2), a solution is given by
Q2=A%*%ginv(X%*%A)%*%W
I do not know how to create a matrix Q that satisfies the two conditions. I have try to construct an iterative procedure without success (no convergence):
eps=10
i=0
while(eps>.5)
{
Q1=svd(Q2)$u%*%chol(Z)
Q2=Q1%*%ginv(X%*%Q1)%*%W
eps=sum(abs(Q1-Q2))
cat(i,":",eps,"\n")
i=i+1
}Perhaps someone could have any idea to solve the problem, or a reference on this kind of question or the email of another list where I should ask this question.
Thanks in advance,
Sincerely.
Stéphane DRAY
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