Some algebraic identities for the alpha-permanent

TL;DR

This paper introduces algebraic identities linking the alpha-permanent to determinants and immanants, providing new formulas and discussing computational complexity, with numerical examples.

Contribution

It presents novel algebraic identities for the alpha-permanent, including expressing the permanent as a combination of determinants and relating alpha-permanents to other matrix functions.

Findings

Permanent can be expressed as a linear combination of determinants of block matrices.

Alpha-permanent can be written as a combination of related alpha-permanents.

Discussion on the computational complexity of alpha-permanent.

Abstract

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.