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The exception is the non-negative monomial matrices: a non-negative matrix has non-negative inverse if and only if it is a (non-negative) monomial matrix. Note that thus the inverse of a positive matrix is not positive or even non-negative, as positive matrices are not monomial, for dimension n > 1.
Non-negative matrix factorization (NMF or NNMF), also non-negative matrix approximation [1] [2] is a group of algorithms in multivariate analysis and linear algebra where a matrix V is factorized into (usually) two matrices W and H, with the property that all three matrices have no negative elements. This non-negativity makes the resulting ...
In mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector, where is the row vector transpose of . [1] More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number is positive for every nonzero complex column vector , where denotes the ...
A matrix whose off-diagonal entries are non-negative. Monomial matrix: A square matrix with exactly one non-zero entry in each row and column. Synonym for generalized permutation matrix. Moore matrix: A row consists of a, a q, a q², etc., and each row uses a different variable. Nonnegative matrix: A matrix with all nonnegative entries. Null ...
The exponential of a Metzler (or quasipositive) matrix is a nonnegative matrix because of the corresponding property for the exponential of a nonnegative matrix. This is natural, once one observes that the generator matrices of continuous-time Markov chains are always Metzler matrices, and that probability distributions are always non-negative.
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 ...
If the diagonal elements of D are real and non-negative then it is positive semidefinite, and if the square roots are taken with the (+) sign (i.e. all non-negative), the resulting matrix is the principal root of D. A diagonal matrix may have additional non-diagonal roots if some entries on the diagonal are equal, as exemplified by the identity ...
The singular values are non-negative real numbers, usually listed in decreasing order (σ 1 (T), σ 2 (T), …). The largest singular value σ 1 (T) is equal to the operator norm of T (see Min-max theorem). Visualization of a singular value decomposition (SVD) of a 2-dimensional, real shearing matrix M.