A permanent matrix

Writing:

pr=perm(matrix);

Returns permanent pr matrix matrix

The matrix permanent is one of the fundamental mathematical functions widely used in various fields of science and engineering. Unlike the determinant, which operates with sums of products of matrix elements with an equal number of terms, the permanent takes into account all possible products of matrix elements, where each product is taken into account only once.

Formally, the permanent of a matrix A of size n x n is defined as the sum of the products of all possible permutations of matrix elements:

perm(A) = ∑Πa_{i, j}π_{i, j}

where pi is a permutation of indices from 1 to n, perm(A) is the sign of the permutation, a[i, j] is an element of the matrix A located in the i-th row and j-th column.

The permanent matrix has several important properties, which makes it a useful tool in various fields of science and technology. First, the permanent is an invariant with respect to the permutation of rows and columns of the matrix. This means that changing the order of rows and columns does not affect the value of the permanent. Secondly, a permanent can be considered as a sum of products of elements, which makes it useful for analyzing combinatorial problems. Thirdly, the permanent can be used to solve optimization problems, for example, to find the maximum permutation transformation of a matrix.

One of the most well-known applications of the matrix permanent is its use in coding theory. In particular, the permanent is used to determine the properties and parameters of Boolean functions, which in turn are used to construct effective codes and correct errors in data transmission. The permanent also finds applications in graph theory, commutative algebra, quantum physics and other fields.

Using a permanent matrix can be quite computationally difficult, as it has exponential complexity for conventional algorithms. However, there are effective approximate algorithms and heuristics for calculating the permanent, which allow us to obtain fairly accurate results for practical tasks.

In conclusion, the matrix permanent is an important mathematical function with wide application in various fields of science and engineering. Its properties and possibilities of use make it an essential tool for the analysis of combinatorial problems, optimization, coding theory and many other fields.