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.
Definition The permanent of an n×n matrix A = (ai,j) is defined as
perm ( A ) = ∑ σ ∈ S n ∏ i = 1 n a i , σ ( i ) . {\displaystyle \operatorname {perm} (A)=\sum _{\sigma \in S_{n}}\prod _{i=1}^{n}a_{i,\sigma (i)}.}
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,
perm ( a b c d ) = a d + b c , {\displaystyle \operatorname {perm} {\begin{pmatrix}a&b\\c&d\end{pmatrix}}=ad+bc,}
and
perm ( a b c d e f g h i ) = a e i + b f g + c d h + c e g + b d i + a f h . {\displaystyle \operatorname {perm} {\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix}}=aei+bfg+cdh+ceg+bdi+afh.}
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 by per A, perm A, or Per A. Minc uses Per(A) for the permanent of a rectangular matrix, and per(A) when A is a square matrix. Muir and Metzler use 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 and Metzler in the more specific modern sense.
Properties 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 A = ( a i j ) {\displaystyle A=\left(a_{ij}\right)} of order n:
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