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When creating a data-set of terms that appear in a corpus of documents, the document-term matrix contains rows corresponding to the documents and columns corresponding to the terms. Each ij cell, then, is the number of times word j occurs in document i. As such, each row is a vector of term counts that represents the content of the document ...
Power Query was first announced in 2011 under the codename "Data Explorer" as part of Azure SQL Labs. In 2013, in order to expand on the self-service business intelligence capabilities of Microsoft Excel, the project was redesigned to be packaged as an add-in Excel and was renamed "Data Explorer Preview for Excel" [4], and was made available for Excel 2010 and Excel 2013. [5]
Typically, the matrix is assumed to be stored in row-major or column-major order (i.e., contiguous rows or columns, respectively, arranged consecutively). Performing an in-place transpose (in-situ transpose) is most difficult when N ≠ M , i.e. for a non-square (rectangular) matrix, where it involves a complex permutation of the data elements ...
The transpose (indicated by T) of any row vector is a column vector, and the transpose of any column vector is a row vector: […] = [] and [] = […]. The set of all row vectors with n entries in a given field (such as the real numbers ) forms an n -dimensional vector space ; similarly, the set of all column vectors with m entries forms an m ...
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix A by producing another matrix, often denoted by A T (among other notations). [1] The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley. [2]
Multiplying a matrix M by either or on either the left or the right will permute either the rows or columns of M by either π or π −1.The details are a bit tricky. To begin with, when we permute the entries of a vector (, …,) by some permutation π, we move the entry of the input vector into the () slot of the output vector.
Multiplication of X by e i extracts the i-th column, while multiplication by B i puts it into the desired position in the final vector. Alternatively, the linear sum can be expressed using the Kronecker product : vec ( X ) = ∑ i = 1 n e i ⊗ X e i {\displaystyle \operatorname {vec} (\mathbf {X} )=\sum _{i=1}^{n}\mathbf {e} _{i}\otimes ...
Decomposition: = where C is an m-by-r full column rank matrix and F is an r-by-n full row rank matrix Comment: The rank factorization can be used to compute the Moore–Penrose pseudoinverse of A , [ 2 ] which one can apply to obtain all solutions of the linear system A x = b {\displaystyle A\mathbf {x} =\mathbf {b} } .