What`s a system of linear equations
... Thm 2. Existence and uniqueness 1. A linear system is consistent if and only if echelon form of the augmented matrix has no row like [0, 0, ….0, b]. Here b ≠ 0 2. A linear system is consistent. Then either (i) it has unique solution when there is no free variables; or (ii) it has infinitely many so ...
... Thm 2. Existence and uniqueness 1. A linear system is consistent if and only if echelon form of the augmented matrix has no row like [0, 0, ….0, b]. Here b ≠ 0 2. A linear system is consistent. Then either (i) it has unique solution when there is no free variables; or (ii) it has infinitely many so ...
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Cayley-Hamilton theorem over a Field
... We look at the set of n equations above, whose coefficients form an n x n matrix. Since F[X] is a Euclidean ring, it is possible by using row operations on that matrix, to change it to an upper-triangular form, for which the corresponding set of n equations is still valid, and such that the determin ...
... We look at the set of n equations above, whose coefficients form an n x n matrix. Since F[X] is a Euclidean ring, it is possible by using row operations on that matrix, to change it to an upper-triangular form, for which the corresponding set of n equations is still valid, and such that the determin ...
Non-negative matrix factorization
NMF redirects here. For the bridge convention, see new minor forcing.Non-negative matrix factorization (NMF), also non-negative matrix approximation 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 matrices easier to inspect. Also, in applications such as processing of audio spectrograms non-negativity is inherent to the data being considered. Since the problem is not exactly solvable in general, it is commonly approximated numerically.NMF finds applications in such fields as computer vision, document clustering, chemometrics, audio signal processing and recommender systems.