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Permanent

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In linear algebra, the permanent of a matrix is a function of a matrix related to the determinant. The permanent as well as the determinant are polynomials of the entries of the matrix.

Definition

The permanent of an n-by-n matrix A = (a_i,j) is defined as

operatorname{perm}(A)=sum_{sigmain S_n}prod_{i=1}^n a_{i,sigma(i)}.

The sum here extends over all elements σ of the symmetric group S_n, i.e. over all permutations of the numbers 1, 2, ..., n.

For example,

operatorname{perm}begin{pmatrix}a&b  c&dend{pmatrix}=ad+bc.

The definition of the permanent of A differs from that of the determinant of A in that the signatures of the permutations are not taken into account. If one views the permanent as a map that takes n vectors as arguments, then it is a multilinear map and it is symmetric (meaning that any order of the vectors results in the same permanent). A formula similar to Laplace's for the development of a determinant along a row or column is also valid for the permanent; all signs have to be ignored for the permanent.

Applications

Unlike the determinant, the permanent has no easy geometrical interpretation; it is mainly used in combinatorics. The permanent describes the number of perfect matchings in a bipartite graph. More specifically, let G be a bipartite graph with vertices A₁, A₂, ..., A_n on one side and B₁, B₂, ..., B_n on the other side. Then, G can be described by an n-by-n matrix A = (a_i,j) where a_i,j = 1 if there is an edge between the vertices A_i and B_j and a_i,j = 0 otherwise. The permanent of this matrix is equal to the number of perfect matchings in the graph.

Complexity

The permanent is also more difficult to compute than the determinant. While the determinant can be computed in polynomial time by Gaussian elimination, Gaussian elimination cannot be used to compute the permanent. Moreover, computing the permanent of a 0-1 matrix (matrix whose entries are 0 or 1) is #P-complete (proof). Thus, if the permanent can be computed in polynomial time by any method, then FP = #P which is an even stronger statement than P = NP. When the entries of A are nonnegative, however, the permanent can be computed approximately in probabilistic polynomial time, up to an error of εM, where M is the value of the permanent and ε > 0 is arbitrary.

Immanant

The permanent and the determinant are both special cases of the immanant: Given a complex character $chi: S_nrightarrowmathbb{C}$ of the symmetric group $S_n$ , the immanant corresponding to $chi$ of an n-by-n matrix A is

operatorname{imm}_chi(A)=sum_{sigmain S_n}chi(sigma)prod_{i=1}^n a_{i,sigma(i)}.

The permanent is recovered from this definition by taking $chi$ to be the trivial character $sigmamapsto 1$ , and the determinant is recovered by taking $chi$ to be the sign function $sgn$ , which is the unique nontrivial one-dimensional irreducible character of $S_n$ .