Permanent
In
linear algebra, the
permanent of an
nby
n 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 number 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. 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.