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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]
Schematic depiction of the matrix product AB of two matrices A and B. Date: 4 October 2010 (original upload date) Source: This file was derived from: Matrix multiplication diagram.svg: Author: File:Matrix multiplication diagram.svg:User:Bilou; See below.
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:
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English: Multiplication of an orthogonal matrix by by the its transpose creates do products among the rows. The product of an orthogonal matrix by its transpose is the identity matrix because Since the rows are an orthonormal set of basis vectors, the product of an orthonormal matrix with its transpose creates the identity matrix.
English: A depiction of block matrix multiplication. Each input matrix is split into a block matrix, with submatrices small enough to fit in fast memory. A single submatrix of the output matrix is formed from a row of submatrices of the first input and a column of submatrices of the second input.
Multiplication of two matrices is defined if and only if the number of columns of the left matrix is the same as the number of rows of the right matrix. That is, if A is an m × n matrix and B is an s × p matrix, then n needs to be equal to s for the matrix product AB to be defined.