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NumPy (pronounced / ˈ n ʌ m p aɪ / NUM-py) is a library for the Python programming language, adding support for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays. [3]
An optimal threshold matrix is one that for any possible quantisation of color has the minimum possible texture so that the greatest impression of the underlying feature comes from the image being quantised. It can be proven that for matrices whose side length is a power of two there is an optimal threshold matrix. [2]
After the algorithm has converged, the singular value decomposition = is recovered as follows: the matrix is the accumulation of Jacobi rotation matrices, the matrix is given by normalising the columns of the transformed matrix , and the singular values are given as the norms of the columns of the transformed matrix .
The matrix left-division operator concisely expresses some semantic properties of matrices. As in the scalar equivalent, if the (determinant of the) coefficient (matrix) A is not null then it is possible to solve the (vectorial) equation A * x = b by left-multiplying both sides by the inverse of A: A −1 (in both MATLAB and GNU Octave ...
In order to use Dantzig–Wolfe decomposition, the constraint matrix of the linear program must have a specific form. A set of constraints must be identified as "connecting", "coupling", or "complicating" constraints wherein many of the variables contained in the constraints have non-zero coefficients.
The conjugate transpose, therefore, arises very naturally as the result of simply transposing such a matrix—when viewed back again as an matrix made up of complex numbers. For an explanation of the notation used here, we begin by representing complex numbers e i θ {\displaystyle e^{i\theta }} as the rotation matrix, that is,
The scalar matrices are the center of the algebra of matrices: that is, they are precisely the matrices that commute with all other square matrices of the same size. [ a ] By contrast, over a field (like the real numbers), a diagonal matrix with all diagonal elements distinct only commutes with diagonal matrices (its centralizer is the set of ...
Programming languages that implement matrices may have easy means for vectorization. In Matlab/GNU Octave a matrix A can be vectorized by A(:). GNU Octave also allows vectorization and half-vectorization with vec(A) and vech(A) respectively. Julia has the vec(A) function as well.