Matrix multiplication
This article gives an overview of the various ways to multiply
matrices.
The Einstein summation convention is used throughout.
By far the most important way to multiply matrices is the usual matrix multiplication. It is defined between two matrices only if the number of columns of the first matrix is the same as the number of rows of the second matrix. If A is an mbyn matrix and B is an nbyp matrix, then their product AB is an mbyp matrix given by
 (AB)_{ij} = A_{ir}B_{rj} = a_{i1} * b_{1j} + a_{i2} * b_{2j} + ... + a_{in} * b_{nj}
for each pair i and j.
The following picture shows how to calculate the (AB)_{12} element of AB if A is a 2x4 matrix, and B is a 4x3 matrix. Elements from each matrix are paired off in the direction of the arrows; each pair is multiplied and the products are added. The location of the resulting number in AB corresponds to the row and column that were considered.


This notion of multiplication is important because A and B are interpreted as linear transformations (which is almost universally done), then the matrix product AB corresponds to the composition of the two linear transformations, with B being applied first. In general, matrix multiplication is not commutative; that is, AB is not equal to BA.
The complexity of matrix multiplication, if carried out naively, is O(n³), but more efficient algorithms do exist. Strassen's method, devised by Volker Strassen in 1969 and often referred to as "fast matrix multiplication", uses a mapping of bilinear combinations to reduce complexity to O(n^{log2(7)}) (approximately O(n^{2.807...})). In practice, though, it is rarely used since it is awkward to implement, lacking numerical stability. The constant factor involved is about 4.695 asymptotically; Winograd's method improves on this slightly by reducing it to an asymptotic 4.537.
The best algorithm currently known, which was presented by Don Coppersmith and S. Winograd in 1990, has an asymptotic complexity of O(n^{2.376}). It has been shown that the leading exponent must be at least 2.
For two matrices of the same dimensions, we have the Hadamard product or entrywise product. The Hadamard product of two mbyn matrices A and B, denoted by A · B, is an mbyn matrix given by
(A·B)[i,j]=A[i,j] * B[i,j]. For instance
Note that the Hadamard product is a
submatrix of the Kronecker product (see below). Hadamard product is studied by matrix theorists, but it is virtually untouched by linear algebraists.
For any two arbitrary matrices A=(a_{ij}) and B, we have the direct product or Kronecker product A B defined as
(the HTML entity ⊗ (⊗) represents the direct product, but is not supported on older browsers)
Note that if A is mbyn and B is pbyr then A B is an mpbynr matrix. Again this multiplication is not commutative.
For example
 .
If A and B represent linear transformations V_{1} → W_{1} and V_{2} → W_{2}, respectively, then A B represents the tensor product of the two maps,
V_{1} V_{2} →
W_{1} W_{2}.
All three notions of matrix multiplication are associative
 A * (B * C) = (A * B) * C
and
distributive:
 A * (B + C) = A * B + A * C
and
 (A + B) * C = A * C + B * C
and compatible with scalar multiplication:
 c(A * B) = (cA) * B = A * (cB)
Scalar multiplication
The scalar multiplication of a matrix A=(a_{ij}) and a scalar r gives the product
 rA=(r a_{ij}).
If we are concerns with matrices over a ring, then the above multiplicaion
is sometimes called the
left multiplication while the
right multiplication is defined to be
 Ar=(a_{ij} r).
When the underlying ring is
commutative, for example, the real or complex number field, the two multiplications are the same. However, if the ring is not commutative, such as the
quaternions, they may be different. For example
External links
References
 Strassen, Volker, Gaussian Elimination is not Optimal, Numer. Math. 13, p. 354356, 1969
 Coppersmith, D., Winograd S., Matrix multiplication via arithmetic progressions, J. Symbolic Comput. 9, p. 251280, 1990