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Mathematicians please help.
Posted:
Sep 19, 1996 12:14 PM


Hi,
does anyone have any thoughts on how to solve the following problem.
Let 1 J = v' [ F(s)' [ F(s) F(s)'] F(s) ] v
given "F(s)" and "v" find "s" that minimises "J".
1 Jo = min v' [ F(s)' [ F(s) F(s)'] F(s) ] v s
Where "F(s)" is a first order matrix polynomial (see below), "v" is a comlpex nX1 vector and " ' " means conjugate transpose.
F(s) = FA  Fs
Where "F" is a real rXn matrix "A" is a real nXn matrix "s" is a complex scalar and "n" is always greater than "r".
Here are some observations
1) When v and s are real then a solution can be found. By evaluatong "J" and differentiating the resulting rational polynomial.
1 2) The matrix "[ F(s)' [ F(s) F(s)'] F(s) ]" has the following properties
i) It is idempotent ( F(s) F(s) = F(s) ) ii) It is a projector. iii) It has eigenvalues of 0 and 1 only. iv) It is hermtian
2 3) The function always evalutes to a real value between 0 and v 2 where . is the euclidean vector norm.
4) Examples show the function is NOT convex there may be local minima.
T 5) "F(s)' = F(conj(s)) " where "T" is transpose and "conj(s)" is the complex conjugate of s.
If you can point me in the direction of some relevant references or have some suggestions I would be very grateful. This probelm has been teasing me for the past 2 months and I've finally reached an impasse, having exhausted all my ideas.
Thanks
Stuart...



