Matrix Multiplication Operation

Context:
- It can range from being a Matrix-Matrix Multiplication Operation to being a Matrix-Vector Multiplication Operation to being a Matrix-Scalar Multiplication Operation.
- It can be performed by a Matrix Multiplication System (that implements a matrix multiplication algorithm).
- …
Example(s):
- …
Counter-Example(s):
- Matrix Addition.
- Matrix Factorization.
See: Hadamard Product (Matrices), Dot Product, Linear Algrebra Operation, Hadamard Product (Matrices).

References

(Wikipedia, 2023) ⇒ https://en.wikipedia.org/wiki/Matrix_multiplication Retrieved:2023-8-8.
- In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the first and the number of columns of the second matrix. The product of matrices $A$ and $B$ is denoted as $AB$ .
  Matrix multiplication was first described by the French mathematician Jacques Philippe Marie Binet in 1812, to represent the composition of linear maps that are represented by matrices. 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.
  Computing matrix products is a central operation in all computational applications of linear algebra.

(Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Multiplication#Notation_and_terminology Retrieved:2015-1-17.
- In matrix multiplication, there is actually a distinction between the cross and the dot symbols. The cross symbol generally denotes a vector multiplication,Template:Clarification needed while the dot denotes a scalar multiplication. A similar convention distinguishes between the cross product and the dot product of two vectors. …
  … However, matrix multiplication is not commutative ...

(Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Matrix_multiplication#Matrix_product_.28two_matrices.29 Retrieved:2015-1-17.
- In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. Numbers such as the real or complex numbers can be multiplied according to elementary arithmetic. On the other hand, matrices are arrays of numbers, so there is no unique way to define "the" multiplication of matrices. As such, in general the term "matrix multiplication" refers to a number of different ways to multiply matrices. The key features of any matrix multiplication include: the number of rows and columns the original matrices have (called the "size", "order" or "dimension"), and specifying how the entries of the matrices generate the new matrix. ...

(Cohen & Lewis, 1999) ⇒ Edith Cohen, and David D. Lewis. (1999). “Approximating Matrix Multiplication for Pattern Recognition Tasks.” In: Journal of Algorithms, 30(2).
- ABSTRACT: Many pattern recognition tasks, including estimation, classification, and the finding of similar objects, make use of linear models. The fundamental operation in such tasks is the computation of the dot product between a query vector and a large database of instance vectors. Often we are interested primarily in those instance vectors which have high dot products with the query. We present a random sampling based algorithm that enables us to identify, for any given query vector, those instance vectors which have large dot products, while avoiding explicit computation of all dot products. We provide experimental results that demonstrate considerable speedups for text retrieval tasks. Our approximate matrix multiplication algorithm is applicable to products of k ≥ 2 matrices and is of independent interest. Our theoretical and experimental analysis demonstrates that in many scenarios, our method dominates standard matrix multiplication.