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In mathematics, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the ...
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:
This reduces the number of matrix additions and subtractions from 18 to 15. The number of matrix multiplications is still 7, and the asymptotic complexity is the same. [6] The algorithm was further optimised in 2017, [7] reducing the number of matrix additions per step to 12 while maintaining the number of matrix multiplications, and again in ...
This means that, treating the input n×n matrices as block 2 × 2 matrices, the task of multiplying n×n matrices can be reduced to 7 subproblems of multiplying n/2×n/2 matrices. Applying this recursively gives an algorithm needing O ( n log 2 7 ) ≈ O ( n 2.807 ) {\displaystyle O(n^{\log _{2}7})\approx O(n^{2.807})} field operations.
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. If A is an m × n matrix and B is an n × p matrix, then their matrix product AB is the m × p matrix whose entries are given by dot product of the corresponding row of A and the corresponding ...
In computer science, Cannon's algorithm is a distributed algorithm for matrix multiplication for two-dimensional meshes first described in 1969 by Lynn Elliot Cannon. [1] [2]It is especially suitable for computers laid out in an N × N mesh. [3]
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The group of scalar n-by-n matrices over a ring R is the centralizer of the subset of n-by-n matrix units in the set of n-by-n matrices over R. [2] The matrix norm (induced by the same two vector norms) of a matrix unit is equal to 1. When multiplied by another matrix, it isolates a specific row or column in arbitrary position.