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Problem in matrix theory
Posted:
Dec 11, 1996 10:34 AM


Hello,
Does anybody has a solution or reference to the following problem?
Let G be a symmetric and positive definite matrix, block partitioned as
[G_11 G_12 G_13 ... G_1n] [G_21 G_22 .... G_2n] [... ] [... ] [G_n1 G_n2 .... G_nn]
where all blocks are square and equally sized, and G_ii is positive semidefinite for each i (which might be obvious?)
The problem is to minimize
Q(a) = (\sum_{i=1}^n a_i G_ii)^{1}
* (\sum_{i,j} a_i a_j G_ij)
* (\sum_{i=1}^n a_i G_ii)^{1}
over all vectors a={a_i}, i=1,...,n, satisfying a_1+...+a_n=1 and a_i>=0 for each i. The minimization should be done in the sense of "definiteness", i.e. if a* is the optimal vector and a is any other vector, then Q(a)Q(a*) is positive semidefinite.
The middle part of the expression can be viewed as a quadratic form in the matrix blocks, while the outer parts, that are inverted, are linear combinations of the diagonal blocks.
The questions are if a vector a* that is optimal in the sense above exists, and, if so, if there is an algorithm to compute it?
Best wishes,
Tobias Ryden   Tobias RydÃ©n Email: tobias@maths.lth.se Dept. of Mathematical Statistics Tel: int+4646 222 4778 Lund University Fax: int+4646 222 4623  Box 118, S221 00 Lund, Sweden WWW: www.maths.lth.se/matstat



