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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:
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]
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 ...
The first to be discovered was Strassen's algorithm, devised by Volker Strassen in 1969 and often referred to as "fast matrix multiplication". [1] The optimal number of field operations needed to multiply two square n × n matrices up to constant factors is still unknown. This is a major open question in theoretical computer science. As of ...
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.
a ij are 1 if i divides j or if j = 1; otherwise, a ij = 0. A (0, 1)-matrix. Shift matrix: A matrix with ones on the superdiagonal or subdiagonal and zeroes elsewhere. a ij = δ i+1,j or a ij = δ i−1,j: Multiplication by it shifts matrix elements by one position. Zero matrix: A matrix with all entries equal to zero. a ij = 0.
Alabama athletics director Greg Byrne makes a case for the Crimson Tide to make the College Football Playoff despite not winning a conference championship and having three losses.
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 ...