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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:
The problem may be solved using dynamic programming. There are many options because matrix multiplication is associative. In other words, no matter how the product is parenthesized, the result obtained will remain the same. For example, for four matrices A, B, C, and D, there are five possible options:
In linear algebra, the Strassen algorithm, named after Volker Strassen, is an algorithm for matrix multiplication.It is faster than the standard matrix multiplication algorithm for large matrices, with a better asymptotic complexity, although the naive algorithm is often better for smaller matrices.
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.
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 ...
Python is a high-level, general-purpose programming language. Its design philosophy emphasizes code readability with the use of significant indentation. [33] Python is dynamically type-checked and garbage-collected. It supports multiple programming paradigms, including structured (particularly procedural), object-oriented and functional ...
Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.
In programming languages [ edit ] In some programming languages, given a two-argument function f (or a binary operator), the outer product, f , of two one-dimensional arrays, A and B , is a two-dimensional array C such that C[i, j] = f(A[i], B[j]) .