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 permanent and determinant are special cases of a more general function of a matrix called the immanant.

The permanent of an n-by-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 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. In his monograph, Minc (1984) uses Per(A) for the permanent of rectangular matrices, and uses per(A) when A is a square matrix. Muir (1882) uses the notation ${\displaystyle {\overset {+}{|}}\quad {\overset {+}{|}}}$.

The word, permanent, originated with Cauchy in 1812 as “fonctions symétriques permanentes” for a related type of function, and was used by Muir (1882) in the modern, more specific, sense.

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