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.