Changes

==Etymology==

==Etymology==

[http://nordan.daynal.org/wiki/index.php?title=English#ca._1100-1500_.09THE_MIDDLE_ENGLISH_PERIOD Middle English], from Anglo-French parmanant, from [[Latin]] permanent-, permanens, present participle of permanēre  to [[endure]], from per- throughout + manēre to remain  

[https://nordan.daynal.org/wiki/index.php?title=English#ca._1100-1500_.09THE_MIDDLE_ENGLISH_PERIOD Middle English], from Anglo-French parmanant, from [[Latin]] permanent-, permanens, present participle of permanēre  to [[endure]], from per- throughout + manēre to remain  

*Date: [http://www.wikipedia.org/wiki/15th_Century 15th century]

*Date: [https://www.wikipedia.org/wiki/15th_Century 15th century]

==Definition==

==Definition==

*continuing or [[enduring]] without [[fundamental]] or marked [[change]] : [[stable]]

*continuing or [[enduring]] without [[fundamental]] or marked [[change]] : [[stable]]

==Description==

==Description==

In [http://en.wikipedia.org/wiki/Linear_algebra linear algebra], the '''permanent''' of a [http://en.wikipedia.org/wiki/Square_matrix square matrix] is a [[function]] of the matrix similar to the [http://en.wikipedia.org/wiki/Determinant 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 [http://en.wikipedia.org/wiki/Immanant immanant].

In [https://en.wikipedia.org/wiki/Linear_algebra linear algebra], the '''permanent''' of a [https://en.wikipedia.org/wiki/Square_matrix square matrix] is a [[function]] of the matrix similar to the [https://en.wikipedia.org/wiki/Determinant 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 [https://en.wikipedia.org/wiki/Immanant immanant].

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

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

:[[File:Perm_2.jpg]]

:[[File:Perm_2.jpg]]

The definition of the permanent of A [[differs]] from that of the determinant of A in that the signatures of the [http://en.wikipedia.org/wiki/Permutations permutations] are not taken into account. If one [[views]] the permanent as a map that takes n [http://en.wikipedia.org/wiki/Euclidean_vector 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 [http://en.wikipedia.org/wiki/Expansion_by_minors 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.

The definition of the permanent of A [[differs]] from that of the determinant of A in that the signatures of the [https://en.wikipedia.org/wiki/Permutations permutations] are not taken into account. If one [[views]] the permanent as a map that takes n [https://en.wikipedia.org/wiki/Euclidean_vector 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 [https://en.wikipedia.org/wiki/Expansion_by_minors 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.

The permanent is [[believed]] to be more [[difficult]] to compute than the determinant. While the determinant can be computed in polynomial time by [http://en.wikipedia.org/wiki/Gaussian_elimination 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 in [[probabilistic]] polynomial time, up to an error of εM, where M is the [[value]] of the permanent and ε > 0 is [[arbitrary]].

The permanent is [[believed]] to be more [[difficult]] to compute than the determinant. While the determinant can be computed in polynomial time by [https://en.wikipedia.org/wiki/Gaussian_elimination 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 in [[probabilistic]] polynomial time, up to an error of εM, where M is the [[value]] of the permanent and ε > 0 is [[arbitrary]].

==External links==

==External links==

* Permanent at [http://planetmath.org/encyclopedia/Permanent.html PlanetMath]

* Permanent at [https://planetmath.org/encyclopedia/Permanent.html PlanetMath]

[[Category: General Reference]]

[[Category: General Reference]]

[[Category: Mathematics]]

[[Category: Mathematics]]