Re: Matrix Minimax, sorta

In article <dj572n$9bs$2@xxxxxxxxxxxxxxxxxxxxxxxxxx>,
Hauke Reddmann <fc3a501@xxxxxxxxxxxxxx> wrote:
>I have the equation aA+bB+cC+...=min.,
>where a,b,c... are scalars, A,B,C are matrices,
>a=b=c=...=0 is ruled out and "min." isn't
>Euclidean norm but the rank.
>
>For special cases (e.g. two 2x2 matrices - just
>fix a/b so that the determinant vanishes) this
>is easy, but can you give a general approach?

The rank is the size of the largest nonsingular
square submatrix. For the rank to be <= r, all
the (r+1) x (r+1) submatrices must have
determinant 0. Those determinants are polynomials
in a,b,c,... So it's essentially reduced to
solving a (maybe rather large) set of polynomials.
Groebner basis techniques may help.

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada



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