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Formally, a parity check matrix H of a linear code C is a generator matrix of the dual code, C ⊥. This means that a codeword c is in C if and only if the matrix-vector product Hc ⊤ = 0 (some authors [1] would write this in an equivalent form, cH ⊤ = 0.) The rows of a parity check matrix are the coefficients of the parity check equations. [2]
The high rank matrix completion in general is NP-Hard. However, with certain assumptions, some incomplete high rank matrix or even full rank matrix can be completed. Eriksson, Balzano and Nowak [10] have considered the problem of completing a matrix with the assumption that the columns of the matrix belong to a union of multiple low-rank subspaces.
As a result, once reduction is complete the parity errors sometimes seen on the 4×4×4 cannot occur on the 5×5×5, or any cube with an odd number of layers. [9] The Yau5 method is named after its proposer, Robert Yau. The method starts by solving the opposite centers (preferably white and yellow), then solving three cross edges (preferably ...
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...
A vector treated as an array of numbers by writing as a row vector or column vector (whichever is used depends on convenience or context): = (), = Index notation allows indication of the elements of the array by simply writing a i, where the index i is known to run from 1 to n, because of n-dimensions. [1]
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced / ʃ ə ˈ l ɛ s k i / shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations.
A parity plot is a scatterplot that compares a set of results from a computational model against benchmark data. Each point has coordinates ( x , y ), where x is a benchmark value and y is the corresponding value from the model.
In matrix theory and combinatorics, a Pascal matrix is a matrix (possibly infinite) containing the binomial coefficients as its elements. It is thus an encoding of Pascal's triangle in matrix form. There are three natural ways to achieve this: as a lower-triangular matrix , an upper-triangular matrix , or a symmetric matrix .
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