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In linear algebra, the nonnegative rank of a nonnegative matrix is a concept similar to the usual linear rank of a real matrix, but adding the requirement that certain coefficients and entries of vectors/matrices have to be nonnegative. For example, the linear rank of a matrix is the smallest number of vectors, such that every column of the matrix can be written as a linear combination of those vectors. For the nonnegative rank, it is required that the vectors must have nonnegative entries, and also that the coefficients in the linear combinations are nonnegative. == Formal Definition == There are several equivalent definitions, all modifying the definition of the linear rank slightly. Apart from the definition given above, there is the following: The nonnegative rank of a nonnegative ''m×n''-matrix ''A'' is equal to the smallest number ''q'' such there exists a nonnegative ''m×q''-matrix ''B'' and a nonnegative ''q×n''-matrix ''C'' such that ''A = BC'' (the usual matrix product). To obtain the linear rank, drop the condition that ''B'' and ''C'' must be nonnegative. Further, the nonnegative rank is the smallest number of nonnegative rank-one matrices into which the matrix can be decomposed additively: where ''Rj ≥ 0'' stands for "''Rj'' is nonnegative".〔Abraham Berman and Robert J. Plemmons. ''Nonnegative Matrices in the Mathematical Sciences'', SIAM〕 (To obtain the usual linear rank, drop the condition that the ''Rj'' have to be nonnegative.) Given a nonnegative matrix ''A'' the nonnegative rank of ''A'' satisfies where denotes the usual linear rank of ''A''. 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「Nonnegative rank (linear algebra)」の詳細全文を読む スポンサード リンク
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