Permanent (mathematics)

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 are special cases of a more general function of a matrix called the immanant. The permanent of an n×n matrix A = (ai,j) 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. For example, and 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. The permanent of a matrix A is denoted per A, perm A, or Per A, sometimes with parentheses around the argument. Minc uses Per(A) for the permanent of rectangular matrices, and per(A) when A is a square matrix. Muir and Metzler use the notation . The word, permanent, originated with Cauchy in 1812 as “fonctions symétriques permanentes” for a related type of function, and was used by Muir and Metzler in the modern, more specific, sense. 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). Furthermore, given a square matrix of order n: perm(A) is invariant under arbitrary permutations of the rows and/or columns of A. This property may be written symbolically as perm(A) = perm(PAQ) for any appropriately sized permutation matrices P and Q, multiplying any single row or column of A by a scalar s changes perm(A) to s⋅perm(A), perm(A) is invariant under transposition, that is, perm(A) = perm(AT). If and are square matrices of order n then, where s and t are subsets of the same size of {1,2,...,n} and are their respective complements in that set. If is a triangular matrix, i.e. , whenever or, alternatively, whenever , then its permanent (and determinant as well) equals the product of the diagonal entries: Laplace's expansion by minors for computing the determinant along a row, column or diagonal extends to the permanent by ignoring all signs.