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linalg::permanent

Permanent of a matrix

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Syntax

linalg::permanent(A)

Description

linalg::permanent(A) computes the permanent of the square matrix A.

The component ring of the matrix A must be a commutative ring, i.e., a domain of category Cat::CommutativeRing.

Examples

Example 1

We compute the permanent of the following matrix:

delete a11, a12, a21, a22:
A := matrix([[a11, a12], [a21, a22]])

which gives us the general formula for the permanent of an arbitrary 2 ×2 matrix:

linalg::permanent(A)

Example 2

The permanent of a matrix can be computed over arbitrary commutative rings. Let us create a random matrix defined over the ring 6, the integers modulo 6:

B := linalg::randomMatrix(5, 5, Dom::IntegerMod(6))

The permanent of this matrix is:

linalg::permanent(B)

Its determinant is:

det(B)

Parameters

A

A square matrix of a domain of category Cat::Matrix

Return Values

Element of the component ring of A.

Algorithms

The permanent of an n×n matrix A = (ai, j)1 ≤ in, 1 ≤ jn is defined similary as the determinant of A, only the signs of the permutations do not enter the definition:

.

(Sn is the symmetric group of all permutations of {1, …, n}.)

In contrast to the computation of the determinant, the computation of the permanent takes time O(n2 2n).

See Also

MuPAD Functions

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