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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: This is different from the identity matrix under regular matrix multiplication, where only the elements of the main diagonal are equal to 1. 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. The identity matrix under hadamard multiplication of two m × n matrices is an m × n matrix where all elements are equal to 1 Strassen algorithm 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 ( versus ), although the naive algorithm is often better for smaller matrices.

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

The runtime of the algorithm is , where is the time of the initial distribution of. It defines the function apd which returns a matrix with entries such that is the length of the shortest path between the vertices and The matrix class used can be any matrix class implementation supporting the multiplication, exponentiation, and indexing operators (for example numpy. Matrix multiplication is thus a basic tool of linear algebra, and as such has numerous applications in many areas of mathematics, as well as in applied mathematics, statistics, physics, economics, and engineering

[3][4] computing matrix products is a central operation in all computational applications of linear algebra.

Matrix chain multiplication 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 following tables provide a comparison of linear algebra software libraries, either specialized or general purpose libraries with significant linear algebra coverage.

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