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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]
Matrix chain multiplication (or the matrix chain ordering problem [1]) is an optimization problem concerning the most efficient way to multiply a given sequence of matrices. The problem is not actually to perform the multiplications, but merely to decide the sequence of the matrix multiplications involved.
The logical operation and takes the place of multiplication. The outer product of two logical vectors (u i) and (v j) is given by the logical matrix () = (). This type of matrix is used in the study of binary relations, and is called a rectangular relation or a cross-vector. [12]
The matrix multiplication exponent, usually denoted ω, is the smallest real number for which any two matrices over a field can be multiplied together using + field operations. This notation is commonly used in algorithms research, so that algorithms using matrix multiplication as a subroutine have bounds on running time that can update as ...
Here, the traditional BLAS functions provide typically good performance for large matrices. However, when computing e.g., matrix-matrix-products of many small matrices by using the GEMM routine, those architectures show significant performance losses. To address this issue, in 2017 a batched version of the BLAS function has been specified. [52]
The left column visualizes the calculations necessary to determine the result of a 2x2 matrix multiplication. Naïve matrix multiplication requires one multiplication for each "1" of the left column. Each of the other columns (M1-M7) represents a single one of the 7 multiplications in the Strassen algorithm. The sum of the columns M1-M7 gives ...
The Hadamard product operates on identically shaped matrices and produces a third matrix of the same dimensions. In mathematics, the Hadamard product (also known as the element-wise product, entrywise product [1]: ch. 5 or Schur product [2]) is a binary operation that takes in two matrices of the same dimensions and returns a matrix of the multiplied corresponding elements.