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Replies: 0

 SJ Griffin Posts: 4 Registered: 12/7/04
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...