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.
The permanent of an n
) is defined as
The sum here extends over all elements σ of the symmetric group Sn, i.e. over all permutations of the numbers 1, 2, ..., n.
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
. The permanent describes the number of perfect matchings
in a bipartite graph
. More specifically, let G
be a bipartite graph with vertices A1
, ..., An
on one side and B1
, ..., Bn
on the other side. Then, G
can be described by an n
) where ai,j
= 1 if there is an edge
between the vertices Ai
= 0 otherwise. The permanent of this matrix is equal to the number of perfect matchings in the graph.
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
). 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
polynomial time, up to an error of εM
, where M
is the value of the permanent and ε > 0 is arbitrary. Because the permanent is random self-reducible
, these results hold out even for average-case
The permanent and the determinant are both special cases of the immanant: Given a complex character of the symmetric group , the immanant corresponding to of an n-by-n matrix A is
The permanent is recovered from this definition by taking to be the trivial character , and the determinant is recovered by taking to be the sign function , which is the unique nontrivial one-dimensional irreducible character of .