  To apportion a complex matrix means to apply a similarity so that all entries
of the resulting matrix have the same magnitude. We initiate the study of
apportionment, both by unitary matrix similarity and general matrix similarity.
There are connections between apportionment and classical graph decomposition
problems, including graceful labelings of graphs, Hadamard matrices, and
equiangluar lines, and potential applications to instantaneous uniform mixing
in quantum walks. The connection between apportionment and graceful labelings
allows the construction of apportionable matrices from trees. A generalization
of the well-known Eigenvalue Interlacing Inequalities using graceful labelings
is also presented. It is shown that every rank one matrix can be apportioned by
a unitary similarity, but there are $2\x 2$ matrices that cannot be
apportioned. A necessary condition for a matrix to be apportioned by unitary
matrix is established. This condition is used to construct a set of matrices
with nonzero Lebesgue measure that are not apportionable by a unitary matrix.
