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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:
In practice we have much fewer processors than the matrix elements. We can replace the matrix elements with submatrices, so that every processor processes more values. The scalar multiplication and addition become sequential matrix multiplication and addition. The width and height of the submatrices will be = /.
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]
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 ...
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 ...
Freivalds' algorithm (named after Rūsiņš Mārtiņš Freivalds) is a probabilistic randomized algorithm used to verify matrix multiplication. Given three n × n matrices A {\displaystyle A} , B {\displaystyle B} , and C {\displaystyle C} , a general problem is to verify whether A × B = C {\displaystyle A\times B=C} .
Download as PDF; Printable version; In other projects Wikidata item; Appearance. move to sidebar hide. Help ... Pages in category "Matrix multiplication algorithms"
The straightforward multiplication of a matrix that is X × Y by a matrix that is Y × Z requires XYZ ordinary multiplications and X(Y − 1)Z ordinary additions. In this context, it is typical to use the number of ordinary multiplications as a measure of the runtime complexity. If A is a 10 × 30 matrix, B is a 30 × 5 matrix, and C is a 5 × ...