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The iterative proportional fitting procedure (IPF or IPFP, also known as biproportional fitting or biproportion in statistics or economics (input-output analysis, etc.), RAS algorithm [1] in economics, raking in survey statistics, and matrix scaling in computer science) is the operation of finding the fitted matrix which is the closest to an initial matrix but with the row and column totals of ...
rank(A) = the maximum number of linearly independent rows or columns of A. [5] If the matrix represents a linear transformation, the column space of the matrix equals the image of this linear transformation. The column space of a matrix A is the set of all linear combinations of the columns in A. If A = [a 1 ⋯ a n], then colsp(A) = span({a 1 ...
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.
A person must have their own birth certificate, it is specific to that person by its Id number. One-to-one (optional on one side) person ←→ driving license: 1: 0..1 or ? A person may have a driving license, it is specific to that person by its Id number. One-to-many: order ←→ line item: 1: 1..* or + An order contains at least one item ...
Multiplying a row by a number multiplies the determinant by this number. Adding a multiple of one row to another row does not change the determinant. The above properties relating to rows (properties 2–4) may be replaced by the corresponding statements with respect to columns. The determinant is invariant under matrix similarity.
More generally, there are d! possible orders for a given array, one for each permutation of dimensions (with row-major and column-order just 2 special cases), although the lists of stride values are not necessarily permutations of each other, e.g., in the 2-by-3 example above, the strides are (3,1) for row-major and (1,2) for column-major.
The all-ones matrix arises in the mathematical field of combinatorics, particularly involving the application of algebraic methods to graph theory.For example, if A is the adjacency matrix of an n-vertex undirected graph G, and J is the all-ones matrix of the same dimension, then G is a regular graph if and only if AJ = JA. [7]
In an EAV data model, each attribute–value pair is a fact describing an entity, and a row in an EAV table stores a single fact. EAV tables are often described as "long and skinny": "long" refers to the number of rows, "skinny" to the few columns. Data is recorded as three columns: The entity: the item being described.