Least square optimization (of matrices) in R

Yesterday I asked a question about least square optimization in R and it turned out that lm function is the thing that I was looking for.

On the other hand, now I have an other least square optimization question and I am wondering if lm could also solve this problem, or if not, how it can be handled in R.

I have fixed matrices B (of dimension n x m) and V (of dimension n x n), I am looking for an m-long vector u such that

       sum( ( V - ( B %*% diag(u) %*% t(B)) )^2 )

is minimized.

Solution

1) lm.fit Use the fact that

vec(AXA') = (A ⊗ A ) vec(X)

so:

k <- ncol(A)
AA1 <- kronecker(A, A)[, c(diag(k)) == 1]
lm.fit(AA1, c(V))

Here is a self contained example:

# test data
set.seed(123)
A <- as.matrix(BOD)
u <- 1:2
V <- A %*% diag(u) %*% t(A) + rnorm(36)

# solve
k <- ncol(A)
AA1 <- kronecker(A, A)[, c(diag(k)) == 1]
fm1 <- lm.fit(AA1, c(V))

giving roughly the original coefficients 1:2 :

> coef(fm1)
      x1       x2 
1.011206 1.999575

2) nls We can alternately use nls like this:

fm2 <- nls(c(V) ~ c(A %*% diag(x) %*% t(A)), start = list(x = numeric(k)))

giving the following for the above example:

> fm2
Nonlinear regression model
  model: c(V) ~ c(A %*% diag(x) %*% t(A))
   data: parent.frame()
   x1    x2 
1.011 2.000 
 residual sum-of-squares: 30.52

Number of iterations to convergence: 1 
Achieved convergence tolerance: 1.741e-09

Update: Corrections and second solution.