Let f = det(X'X)^{-m/2}  * etr ( -Y (X'X)^{-1}  Y').

Here both X and Y are real matrices. 
X has full column rank. 
X' denotes the matrix transpose.
X^{-1} is the matrix inverse. 
det(.) denotes the determinant of a matrix.
etr(.) is the exponential trace function etr(.) = exp( trace(.) ).  
m is a constant scalar. 
Y is a constant matrix. 

I am interested in finding X that maximizes the function f. 

1. If we assume we know nothing about X, and also assume Y has full
column rank, then function f has maximum. Dr. Israel has solved this
problem. See http://groups.google.com/group/sci.math/msg/742abcca5e9998ec

2. Now the problem is what if we know some columns of X?  That is, if
some columns of X are known to be fixed, how to find other unknown
columns to maximize the function f?

Thank you very much.

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Clarification of Question by elgoog_elgoog-ga on 18 Oct 2006 18:35 PDT

For the sake of simplicity, you can assume all columns of X are known
except the first column. Thus the problem has been reduced to finding
the first column of X that maximizes the function.

Thanks