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Some algebraic identities for the alpha-permanent

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 α-permanents: for arbitrary complex numbers αand β, we show that the α-permanent of any matrix can be expressed as a linear combination of β-permanents of related matrices. Some other identities for the α-permanent of sums and products of matrices are shown, as well as a relationship between the α-permanent and general immanants. We conclude with a discussion of the computational complexity of the α-permanent and provide some numerical illustrations.