  We show that the permanent of a matrix is a linear combination of
determinants of block diagonal matrices which are simple functions of the
original matrix. To prove this, we first show a more general identity involving
\alpha-permanents: for arbitrary complex numbers \alpha and \beta, we show that
the \alpha-permanent of any matrix can be expressed as a linear combination of
\beta-permanents of related matrices. Some other identities for the
\alpha-permanent of sums and products of matrices are shown, as well as a
relationship between the \alpha-permanent and general immanants. We conclude
with a discussion of the computational complexity of the \alpha-permanent and
provide some numerical illustrations.
