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In 1926, Van der Waerden conjectured that the minimum permanent among all n × n doubly stochastic matrices is n!/n n, achieved by the matrix for which all entries are equal to 1/n. [18] Proofs of this conjecture were published in 1980 by B. Gyires [ 19 ] and in 1981 by G. P. Egorychev [ 20 ] and D. I. Falikman; [ 21 ] Egorychev's proof is an ...
Any square matrix = can be viewed as the adjacency matrix of a directed graph, with representing the weight of the edge from vertex to vertex .Then, the permanent of is equal to the sum of the weights of all cycle-covers of the graph; this is a graph-theoretic interpretation of the permanent.
While the compression operator maps the class of 1-semi-unitary matrices to itself and the classes of unitary and 2-semi-unitary ones, the compression-closure of the 1-semi-unitary class (as well as the class of matrices received from unitary ones through replacing one row by an arbitrary row vector — the permanent of such a matrix is, via ...
Retrieved from "https://en.wikipedia.org/w/index.php?title=Computation_of_the_permanent_of_a_matrix&oldid=260146503"
Hadamard's maximal determinant problem, named after Jacques Hadamard, asks for the largest determinant of a matrix with elements equal to 1 or −1. The analogous question for matrices with elements equal to 0 or 1 is equivalent since, as will be shown below, the maximal determinant of a {1,−1} matrix of size n is 2 n−1 times the maximal determinant of a {0,1} matrix of size n−1.
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. [1]If A is a differentiable map from the real numbers to n × n matrices, then