Matrix multiplication

This article gives an overview of the various ways to multiply matrices.

The Einstein summation convention is used throughout.

Table of contents

1 Ordinary matrix product

1.1 Example

2 Hadamard product
3 Kronecker product
4 Common properties
5 Scalar multiplication
6 External links
7 References

Ordinary matrix product

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 m-by-n matrix and B is an n-by-p matrix, then their product AB is an m-by-p matrix given by

(AB)_ij = A_irB_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.

Example

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.

Hadamard product

For two matrices of the same dimensions, we have the Hadamard product or entrywise product. The Hadamard product of two m-by-n matrices A and B, denoted by A · B, is an m-by-n 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.

Kronecker product

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 ⊗ (&otimes;) represents the direct product, but is not supported on older browsers)

Note that if A is m-by-n and B is p-by-r then A B is an mp-by-nr matrix. Again this multiplication is not commutative.

For example

If A and B represent linear transformations V₁ → W₁ and V₂ → W₂, respectively, then A B represents the tensor product of the two maps, V₁ V₂ → W₁ W₂.

Common properties

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. 354-356, 1969
Coppersmith, D., Winograd S., Matrix multiplication via arithmetic progressions, J. Symbolic Comput. 9, p. 251-280, 1990