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Matrix multiplication shares some properties with usual multiplication. However, matrix multiplication is not defined if the number of columns of the first factor differs from the number of rows of the second factor, and it is non-commutative, [10] even when the product remains defined after changing the order of the factors. [11] [12]
An m × n (read as m by n) order matrix is a set of numbers arranged in m rows and n columns. Matrices of the same order can be added by adding the corresponding elements. Two matrices can be multiplied, the condition being that the number of columns of the first matrix is equal to the number of rows of the second matrix.
This was really only relevant for presentation, because matrix multiplication was stack-based and could still be interpreted as post-multiplication, but, worse, reality leaked through the C-based API because individual elements would be accessed as M[vector][coordinate] or, effectively, M[column][row], which unfortunately muddled the convention ...
In theoretical computer science, the computational complexity of matrix multiplication dictates how quickly the operation of matrix multiplication can be performed. Matrix multiplication algorithms are a central subroutine in theoretical and numerical algorithms for numerical linear algebra and optimization, so finding the fastest algorithm for matrix multiplication is of major practical ...
Left multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while right multiplication (post-multiplication) represents elementary column operations. Elementary row operations are used in Gaussian elimination to reduce a matrix to row echelon form .
In linear algebra, a column vector with elements is an matrix [1] consisting of a single column of entries, for example, = [].. Similarly, a row vector is a matrix for some , consisting of a single row of entries, = […]. (Throughout this article, boldface is used for both row and column vectors.)
The definition of matrix multiplication is that if C = AB for an n × m matrix A and an m × p matrix B, then C is an n × p matrix with entries = =. From this, a simple algorithm can be constructed which loops over the indices i from 1 through n and j from 1 through p, computing the above using a nested loop:
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.