  The problem of membrane topology in the matrix model of M-theory is
considered. The matrix regularization procedure, which makes a correspondence
between finite-sized matrices and functions defined on a two-dimensional base
space, is reexamined. It is found that the information of topology of the base
space manifests itself in the eigenvalue distribution of a single matrix. The
precise manner of the manifestation is described. The set of all eigenvalues
can be decomposed into subsets whose members increase smoothly, provided that
the fundamental approximations in matrix regularization hold well. Those
subsets are termed as eigenvalue sequences. The eigenvalue sequences exhibit a
branching phenomenon which reflects Morse-theoretic information of topology.
  Furthermore, exploiting the notion of eigenvalue sequences, a new
correspondence rule between matrices and functions is constructed. The new rule
identifies the matrix elements directly with Fourier components of the
corresponding function, evaluated along certain orbits. The rule has
semi-locality in the base space, so that it can be used for all membrane
topologies in a unified way. A few numerical examples are studied, and
consistency with previously known correspondence rules is discussed.
