In linear algebra
, the permanent
of an n
) is defined as
The sum here extends over all elements σ of the symmetric group
, i.e. over all permutations
of the number 1,2,...,n
The definition of the permanent of A
differs from that of the determinant
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 is not multiplicative. It is also not possible to use Gaussian elimination to compute the permanent; no fast algorithms for its computation are known.