In linear algebra, the permanent of a square matrix is a function of the matrix similar to the determinant. The permanent, as well as the determinant, is a polynomial in the entries of the matrix. Both permanent and determinant are special cases of a more general function of a matrix called the immanant.
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The permanent of an nbyn matrix A = (a_{i,j}) is defined as
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,
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.
Unlike the determinant, the permanent has no easy geometrical interpretation; it is mainly used in combinatorics and in treating boson Green's functions in quantum field theory. However, it has two graphtheoretic interpretations: as the sum of weights of cycle covers of a directed graph, and as the sum of weights of perfect matchings in a bipartite graph.
Any square matrix A = (a_{ij}) can be viewed as the adjacency matrix of a directed graph, with a_{ij} representing the weight of the edge from vertex i to vertex j. A cycle cover of a weighted directed graph is a collection of vertexdisjoint directed cycles in the graph that covers all nodes in the graph. Thus, each vertex i in the graph has a unique "successor" σ(i) in the cycle cover, and σ is a permutation on where n is the number of vertices in the graph. Conversely, any permutation σ on corresponds to a cycle cover in which there is an edge from vertex i to vertex σ(i) for each i.
If the weight of a cyclecover is defined to be the product of the weights of the edges in each cycle, then
The permanent of an matrix A is defined as
where σ is a permutation over . Thus the permanent of A is equal to the sum of the weights of all cyclecovers of the graph.
A square matrix A = (a_{ij}) can also be viewed as the biadjacency matrix of a bipartite graph which has vertices on one side and on the other side, with a_{ij} representing the weight of the edge from vertex x_{i} to vertex y_{j}. If the weight of a perfect matching σ that matches x_{i} to y_{σ(i)} is defined to be the product of the weights of the edges in the matching, then
Thus the permanent of A is equal to the sum of the weights of all perfect matchings of the graph.
In an unweighted, directed, simple graph, if we set each a_{ij} to be 1 if there is an edge from vertex i to vertex j, then each nonzero cycle cover has weight 1, and the adjacency matrix has 01 entries. Thus the permanent of a 01matrix is equal to the number of cyclecovers of an unweighted directed graph.
For an unweighted bipartite graph, if we set a_{i,j} = 1 if there is an edge between the vertices x_{i} and y_{j} and a_{i,j} = 0 otherwise, then each perfect matching has weight 1. Thus the number of perfect matchings in G is equal to the permanent of matrix A.^{[1]}
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 01 matrix (matrix whose entries are 0 or 1) is #Pcomplete. 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.
Permanent used as an adjective usually means that something is very hard to get rid of or destroy. Permanent may also mean other things.
