PCA problem in R

From: Dennis Shea

[SNIP]>> 

On Sat, 13 Aug 2005, Alan Zhao wrote:

When I have more variables than units, say a 195*10896 

matrix which has

10896 variables and 195 samples. prcomp will give only 

195 principal

components. I checked in the help, but there is no 

explanation that why

this happen.

[SNIP]

Sincerely,
Zheng Zhao
Aug-14-2005

Just yesterday I subscribed to r-help because I am planning
on learning the basics of R ... today.   :-)
Thus, I am not sure about the history of this question.

The above situation, more variables than samples, 
is commonly encounterd in the climate studies.
Consider annual mean temperatures for 195 years
on a coarse 72 [lat] x 144 [lon]  grid [72*144=10368 
spatial variables]. 

Let  S be the number of grid points and T be the number
of years. I think there is a theorem (?Eckart-Young?) 
which states that the maximum number of unique eigenvalues 
is min(S,T). In your case 195 eigenvalues is correct. 
I speculate that the underlying function transposes the 
input data matrix and computes the the TxT [rather than SxS]
covariance matrix and solves for the eigenvalues/vectors. 
It then uses a linear transformation to get the results
for the original input data matrix.

Computationally, the above is much faster and uses less memory.