  We develop a new renormalization group approach to the large-N limit of
matrix models. It has been proposed that a procedure, in which a matrix model
of size (N-1) \times (N-1) is obtained by integrating out one row and column of
an N \times N matrix model, can be regarded as a renormalization group and that
its fixed point reveals critical behavior in the large-N limit. We instead
utilize the fuzzy sphere structure based on which we construct a new map
(renormalization group) from N \times N matrix model to that of rank N-1. Our
renormalization group has great advantage of being a nice analog of the
standard renormalization group in field theory. It is naturally endowed with
the concept of high/low energy, and consequently it is in a sense local and
admits derivative expansions in the space of matrices. In construction we also
find that our renormalization in general generates multi-trace operators, and
that nonplanar diagrams yield a nonlocal operation on a matrix, whose action is
to transport the matrix to the antipode on the sphere. Furthermore the
noncommutativity of the fuzzy sphere is renormalized in our formalism. We then
analyze our renormalization group equation, and Gaussian and nontrivial fixed
points are found. We further clarify how to read off scaling dimensions from
our renormalization group equation. Finally the critical exponent of the model
of two-dimensional gravity based on our formalism is examined.
