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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 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 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.
In mathematics, the dot product or scalar product [note 1] is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used.
Scalar multiplication. The product cA of a number c (also called a scalar in this context) and a matrix A is computed by multiplying every entry of A by c: (), =, This operation is called scalar multiplication, but its result is not named "scalar product" to avoid confusion, since "scalar product" is often used as a synonym for "inner product ...
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.
In linear algebra, the outer product of two coordinate vectors is the matrix whose entries are all products of an element in the first vector with an element in the second vector. If the two coordinate vectors have dimensions n and m , then their outer product is an n × m matrix.
This may be seen by writing the zero vector 0 V as 0 ⋅ 0 V (and similarly for 0 W) and moving the scalar 0 "outside", in front of B, by linearity. The set L(V, W; X) of all bilinear maps is a linear subspace of the space (viz. vector space, module) of all maps from V × W into X. If V, W, X are finite-dimensional, then so is L(V, W; X).