DISTANCES BETWEEN THE GRAPHS OF MATRICES

The graph of a complex m-by-n matrix A is the range of [I A(T)](T) in C-m+n. We find explicit formulae for the distance between the graphs o f two m-by-n matrices with respect to two classes of metrics on the se t of subspaces of C-m+n. The singular values of the contraction g(A, B ) = (I+AA)(-1/2)(I+A*B)(I+B*B)(-1/2) play a key role. We study transf ormations that preserve various distances between graphs, and we give some inequalities between different distance functions.