Some algebraic identities for the α-permanent

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 some discussion, and a conjecture for the computational complexity of the alpha-permanent, and provide some numerical illustrations. (C) 2013 Elsevier Inc. All rights reserved.