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In mathematics, especially in probability and combinatorics, a doubly stochastic matrix (also called bistochastic), is a square matrix of nonnegative real numbers, each of whose rows and columns sums to 1, i.e., :, Thus, a doubly stochastic matrix is both left stochastic and right stochastic. Indeed, any matrix that is both left and right stochastic must be square: if every row sums to one then the sum of all entries in the matrix must be equal to the number of rows, and since the same holds for columns, the number of rows and columns must be equal. ==Birkhoff polytope and Birkhoff–von Neumann theorem== The class of doubly stochastic matrices is a convex polytope known as the Birkhoff polytope . Using the matrix entries as Cartesian coordinates, it lies in an -dimensional affine subspace of -dimensional Euclidean space. defined by independent linear constraints specifying that the row and column sums all equal one. (There are constraints rather than because one of these constraints is dependent, as the sum of the row sums must equal the sum of the column sums.) Moreover, the entries are all constrained to be non-negative and less than or equal to one. The Birkhoff–von Neumann theorem states that this polytope is the convex hull of the set of permutation matrices, and furthermore that the vertices of are precisely the permutation matrices. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「doubly stochastic matrix」の詳細全文を読む スポンサード リンク
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