Distances between the graphs of matrices Original Research Article

The graph of a complex m-by-n matrix A is the range of [I AT]T in Cm+n. We find explicit formulae for the distance between the graphs of two m-by-n matrices with respect to two classes of metrics on the set of subspaces of Cm+n. The singular values of the contraction g(A, B) triple bond; length as m-dash (I+A* A)−1/2(I+A* B)(I+B* B)−1/2 play a key role. We study transformations that preserve various distances between graphs, and we give some inequalities between different distance functions.