  In the high-energy quantum-physics literature one finds statements such as
"matrix algebras converge to the sphere". Earlier I provided a general setting
for understanding such statements, in which the matrix algebras are viewed as
compact quantum metric spaces, and convergence is with respect to a quantum
Gromov-Hausdorff-type distance. More recently I have dealt with corresponding
statements in the literature about vector bundles on spheres and matrix
algebras. But physicists want, even more, to treat structures on spheres (and
other spaces) such as Dirac operators, Yang-Mills functionals, etc., and they
want to approximate these by corresponding structures on matrix algebras. In
preparation for understanding what the Dirac operators should be, we determine
here what the corresponding "cotangent bundles" should be for the matrix
algebras, since it is on them that a "Riemannian metric" must be defined, which
is then the information needed to determine a Dirac operator. (In the physics
literature there are at least 3 inequivalent suggestions for the Dirac
operators.)
