  It is well known that a graph is bipartite if and only if the spectrum of its
adjacency matrix is symmetric. In the present paper, this assertion is
dissected into three separate matrix results of wider scope, which are extended
also to hypermatrices. To this end the concept of bipartiteness is generalized
by a new monotone property of cubical hypermatrices, called odd-colorable
matrices. It is shown that a nonnegative symmetric $r$-matrix $A$ has a
symmetric spectrum if and only if $r$ is even and $A$ is odd-colorable. This
result also solves a problem of Pearson and Zhang about hypergraphs with
symmetric spectrum and disproves a conjecture of Zhou, Sun, Wang, and Bu.
  Separately, similar results are obtained for the $H$-spectram of
hypermatrices.
