# Matrix multiplication

In mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring. The matrix product is designed for representing 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, physics, and engineering.[1][2] In more detail, if A is an n × m matrix and B is an m × p matrix, their matrix product AB is an n × p matrix, in which the m entries across a row of A are multiplied with the m entries down a column of B and summed to produce an entry of AB. When two linear maps are represented by matrices, then the matrix product represents the composition of the two maps.

The definition of matrix product requires that the entries belong to ring, which may be noncommutative, but is a field in most applications. Even in this latter case, matrix product is not commutative in general, although it is associative and is distributive over matrix addition. The identity matrices (which are the square matrices whose all entries are zero, except those of the main diagonal that are all equal to 1) are identity elements of the matrix product. It follows that the n × n matrices over a ring form a ring, which is noncommutative except if n = 1 and the ground ring is commutative.

A square matrix may have a multiplicative inverse, called an inverse matrix. In the common case where the entries belong to a commutative ring r, a matrix has an inverse if and only if its determinant has a multiplicative inverse in r. The determinant of a product of square matrices is the product of the determinants of the factors. The n × n matrices that have an inverse form a group under matrix multiplication, the subgroups of which are called matrix groups. Many classical groups (including all finite groups are isomorphic to matrix groups; this is the starting point of the theory of group representations.

Computing matrix products is a central operation in all computational applications of linear algebra. Its computational complexity is ${\displaystyle O(n^{3})}$ (for n × n matrices) for the basic algorithm (this complexity is ${\displaystyle O(n^{2.373})}$ for the asymptotically fastest known algorithm). This nonlinear complexity makes that matrix product is often the critical part of many algorithms. This is enforced by the fact that many operations on matrices, such as matrix inversion, determinant, solving systems of linear equations, have the same complexity. Therefore various algorithms have been devised for computing products of large matrices, taking into account the architecture of computers (see BLAS, for example).

## Notation

This article will use the following notational conventions: matrices are represented by capital letters in bold, e.g. A, vectors in lowercase bold, e.g. a, and entries of vectors and matrices are italic (since they are numbers from a field), e.g. A and a. Index notation is often the clearest way to express definitions, and is used as standard in the literature. The i, j entry of matrix A is indicated by (A)ij or Aij, whereas a numerical label (not matrix entries) on a collection of matrices is subscripted only, e.g. A1, A2, etc.

## Definition

If A is an n × m matrix and B is an m × p matrix,

${\displaystyle \mathbf {A} ={\begin{pmatrix}a_{11}&a_{12}&\cdots &a_{1m}\\a_{21}&a_{22}&\cdots &a_{2m}\\\vdots &\vdots &\ddots &\vdots \\a_{n1}&a_{n2}&\cdots &a_{nm}\\\end{pmatrix}},\quad \mathbf {B} ={\begin{pmatrix}b_{11}&b_{12}&\cdots &b_{1p}\\b_{21}&b_{22}&\cdots &b_{2p}\\\vdots &\vdots &\ddots &\vdots \\b_{m1}&b_{m2}&\cdots &b_{mp}\\\end{pmatrix}}}$

the matrix product AB (denoted without multiplication signs or dots) is defined to be the n × p matrix[3][4][5][6]

${\displaystyle \mathbf {C} ={\begin{pmatrix}c_{11}&c_{12}&\cdots &c_{1p}\\c_{21}&c_{22}&\cdots &c_{2p}\\\vdots &\vdots &\ddots &\vdots \\c_{n1}&c_{n2}&\cdots &c_{np}\\\end{pmatrix}}}$

such that

${\displaystyle \mathbf {C} _{ij}=a_{i1}b_{1j}+\cdots +a_{im}b_{mj}=\sum _{k=1}^{m}a_{ik}b_{kj},}$

for i = 1, ..., n and j = 1, ..., p.

That is, the entry ${\displaystyle c_{ij}}$ of the product is obtained by multiplying term-by-term the entries of the ith row of A and the jth column of B, and summing these m products. In other words, ${\displaystyle c_{ij}}$ is the dot product of the ith row of A and the jth column.

Thus the product AB is defined only if and only if the number of columns in A equals the number of rows in B, in this case m.

Usually the entries are numbers, but they may be any kind mathematical objects for which an addition and a multiplication are defined, that are associative, and such that the addition is commutative, and the multiplication is distributive with respect to the addition. In particular, the entries may be matrices themselves (see block matrix).

### Illustration

The figure to the right illustrates diagrammatically the product of two matrices A and B, showing how each intersection in the product matrix corresponds to a row of A and a column of B.

${\displaystyle {\overset {4\times 2{\text{ matrix}}}{\begin{bmatrix}{a_{11}}&{a_{12}}\\\cdot &\cdot \\{a_{31}}&{a_{32}}\\\cdot &\cdot \\\end{bmatrix}}}{\overset {2\times 3{\text{ matrix}}}{\begin{bmatrix}\cdot &{b_{12}}&{b_{13}}\\\cdot &{b_{22}}&{b_{23}}\\\end{bmatrix}}}={\overset {4\times 3{\text{ matrix}}}{\begin{bmatrix}\cdot &x_{12}&x_{13}\\\cdot &\cdot &\cdot \\\cdot &x_{32}&x_{33}\\\cdot &\cdot &\cdot \\\end{bmatrix}}}}$

The values at the intersections marked with circles are:

{\displaystyle {\begin{aligned}x_{12}&={a_{11}}{b_{12}}+{a_{12}}{b_{22}}\\x_{33}&={a_{31}}{b_{13}}+{a_{32}}{b_{23}}\end{aligned}}}

## Specific dimensions

### Row vector and column vector

If

${\displaystyle \mathbf {A} ={\begin{pmatrix}a&b&c\end{pmatrix}}\,,\quad \mathbf {B} ={\begin{pmatrix}x\\y\\z\end{pmatrix}}\,,}$

their matrix products are:

${\displaystyle \mathbf {AB} ={\begin{pmatrix}a&b&c\end{pmatrix}}{\begin{pmatrix}x\\y\\z\end{pmatrix}}=ax+by+cz\,,}$

and

${\displaystyle \mathbf {BA} ={\begin{pmatrix}x\\y\\z\end{pmatrix}}{\begin{pmatrix}a&b&c\end{pmatrix}}={\begin{pmatrix}xa&xb&xc\\ya&yb&yc\\za&zb&zc\end{pmatrix}}\,.}$

Note AB and BA are two different matrices: the first is a 1 × 1 matrix while the second is a 3 × 3 matrix. Such expressions occur for real-valued Euclidean vectors in Cartesian coordinates, displayed as row and column matrices, in which case AB is the matrix form of their dot product, while BA the matrix form of their dyadic or tensor product.

### Square matrix and column vector

If

${\displaystyle \mathbf {A} ={\begin{pmatrix}a&b&c\\p&q&r\\u&v&w\end{pmatrix}},\quad \mathbf {B} ={\begin{pmatrix}x\\y\\z\end{pmatrix}}\,,}$

their matrix product is:

${\displaystyle \mathbf {AB} ={\begin{pmatrix}a&b&c\\p&q&r\\u&v&w\end{pmatrix}}{\begin{pmatrix}x\\y\\z\end{pmatrix}}={\begin{pmatrix}ax+by+cz\\px+qy+rz\\ux+vy+wz\end{pmatrix}}\,,}$

however BA is not defined.

The product of a square matrix multiplied by a column matrix arises naturally in linear algebra; for solving linear equations and representing linear transformations. By choosing a, b, c, p, q, r, u, v, w in A appropriately, A can represent a variety of transformations such as rotations, scaling and reflections, shears, of a geometric shape in space.

### Square matrices

If

${\displaystyle \mathbf {A} ={\begin{pmatrix}a&b&c\\p&q&r\\u&v&w\end{pmatrix}},\quad \mathbf {B} ={\begin{pmatrix}\alpha &\beta &\gamma \\\lambda &\mu &\nu \\\rho &\sigma &\tau \\\end{pmatrix}}\,,}$

their matrix products are:

${\displaystyle \mathbf {AB} ={\begin{pmatrix}a&b&c\\p&q&r\\u&v&w\end{pmatrix}}{\begin{pmatrix}\alpha &\beta &\gamma \\\lambda &\mu &\nu \\\rho &\sigma &\tau \\\end{pmatrix}}={\begin{pmatrix}a\alpha +b\lambda +c\rho &a\beta +b\mu +c\sigma &a\gamma +b\nu +c\tau \\p\alpha +q\lambda +r\rho &p\beta +q\mu +r\sigma &p\gamma +q\nu +r\tau \\u\alpha +v\lambda +w\rho &u\beta +v\mu +w\sigma &u\gamma +v\nu +w\tau \end{pmatrix}}\,,}$

and

${\displaystyle \mathbf {BA} ={\begin{pmatrix}\alpha &\beta &\gamma \\\lambda &\mu &\nu \\\rho &\sigma &\tau \\\end{pmatrix}}{\begin{pmatrix}a&b&c\\p&q&r\\u&v&w\end{pmatrix}}={\begin{pmatrix}\alpha a+\beta p+\gamma u&\alpha b+\beta q+\gamma v&\alpha c+\beta r+\gamma w\\\lambda a+\mu p+\nu u&\lambda b+\mu q+\nu v&\lambda c+\mu r+\nu w\\\rho a+\sigma p+\tau u&\rho b+\sigma q+\tau v&\rho c+\sigma r+\tau w\end{pmatrix}}\,.}$

In this case, both products AB and BA are defined, and the entries show that AB and BA are not equal in general. Multiplying square matrices which represent linear transformations corresponds to the composite transformation (see below for details).

### Row vector, square matrix, and column vector

If

${\displaystyle \mathbf {A} ={\begin{pmatrix}a&b&c\end{pmatrix}}\,,\quad \mathbf {B} ={\begin{pmatrix}\alpha &\beta &\gamma \\\lambda &\mu &\nu \\\rho &\sigma &\tau \\\end{pmatrix}}\,,\quad \mathbf {C} ={\begin{pmatrix}x\\y\\z\end{pmatrix}}\,,}$

their matrix product is:

{\displaystyle {\begin{aligned}\mathbf {ABC} &={\begin{pmatrix}a&b&c\end{pmatrix}}\left[{\begin{pmatrix}\alpha &\beta &\gamma \\\lambda &\mu &\nu \\\rho &\sigma &\tau \\\end{pmatrix}}{\begin{pmatrix}x\\y\\z\end{pmatrix}}\right]=\left[{\begin{pmatrix}a&b&c\end{pmatrix}}{\begin{pmatrix}\alpha &\beta &\gamma \\\lambda &\mu &\nu \\\rho &\sigma &\tau \\\end{pmatrix}}\right]{\begin{pmatrix}x\\y\\z\end{pmatrix}}\\&={\begin{pmatrix}a&b&c\end{pmatrix}}{\begin{pmatrix}\alpha x+\beta y+\gamma z\\\lambda x+\mu y+\nu z\\\rho x+\sigma y+\tau z\\\end{pmatrix}}={\begin{pmatrix}a\alpha +b\lambda +c\rho &a\beta +b\mu +c\sigma &a\gamma +b\nu +c\tau \end{pmatrix}}{\begin{pmatrix}x\\y\\z\end{pmatrix}}\\&=a\alpha x+b\lambda x+c\rho x+a\beta y+b\mu y+c\sigma y+a\gamma z+b\nu z+c\tau z\,,\end{aligned}}}

however CBA is not defined. Note that A(BC) = (AB)C, this is one of many general properties listed below. Expressions of the form ABC occur when calculating the inner product of two vectors displayed as row and column vectors in an arbitrary coordinate system, and the metric tensor in these coordinates written as the square matrix.

### Rectangular matrices

If

${\displaystyle \mathbf {A} ={\begin{pmatrix}a&b&c\\x&y&z\end{pmatrix}}\,,\quad \mathbf {B} ={\begin{pmatrix}\alpha &\rho \\\beta &\sigma \\\gamma &\tau \\\end{pmatrix}}\,,}$

their matrix products are:

${\displaystyle \mathbf {A} \mathbf {B} ={\begin{pmatrix}a&b&c\\x&y&z\end{pmatrix}}{\begin{pmatrix}\alpha &\rho \\\beta &\sigma \\\gamma &\tau \\\end{pmatrix}}={\begin{pmatrix}a\alpha +b\beta +c\gamma &a\rho +b\sigma +c\tau \\x\alpha +y\beta +z\gamma &x\rho +y\sigma +z\tau \\\end{pmatrix}}\,,}$

and

${\displaystyle \mathbf {B} \mathbf {A} ={\begin{pmatrix}\alpha &\rho \\\beta &\sigma \\\gamma &\tau \\\end{pmatrix}}{\begin{pmatrix}a&b&c\\x&y&z\end{pmatrix}}={\begin{pmatrix}\alpha a+\rho x&\alpha b+\rho y&\alpha c+\rho z\\\beta a+\sigma x&\beta b+\sigma y&\beta c+\sigma z\\\gamma a+\tau x&\gamma b+\tau y&\gamma c+\tau z\end{pmatrix}}\,.}$

## General properties

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, even when the product remains definite after changing the order of the factors.[7][8]

### Non-commutativity

An operation is commutative if, given two elements A and B such that the product ${\displaystyle \mathbf {A} \mathbf {B} }$ is defined, then ${\displaystyle \mathbf {B} \mathbf {A} }$ is also defined, and ${\displaystyle \mathbf {A} \mathbf {B} =\mathbf {B} \mathbf {A} .}$

If A and B are matrices of respective sizes ${\displaystyle m\times n}$ and ${\displaystyle p\times q}$, then ${\displaystyle \mathbf {A} \mathbf {B} }$ is defined if ${\displaystyle n=p}$, and ${\displaystyle \mathbf {B} \mathbf {A} }$ is defined if ${\displaystyle m=q.}$ Therefore, if one of the products is defined, the other is not defined in general. If ${\displaystyle m=q\neq n=p,}$ the two products are defined, but have different sizes; thus they cannot be equal.

It follows that the equality of the two products makes sense only if ${\displaystyle m=q=n=p,}$ that is if A and B are square matrices of the same size. Even in this case, one has in general

${\displaystyle \mathbf {A} \mathbf {B} \neq \mathbf {B} \mathbf {A} .}$

For example

${\displaystyle {\begin{pmatrix}0&1\\0&0\end{pmatrix}}{\begin{pmatrix}0&0\\1&0\end{pmatrix}}={\begin{pmatrix}1&0\\0&0\end{pmatrix}},}$

and

${\displaystyle {\begin{pmatrix}0&0\\1&0\end{pmatrix}}{\begin{pmatrix}0&1\\0&0\end{pmatrix}}={\begin{pmatrix}0&0\\0&1\end{pmatrix}}.}$

This example may be expanded for showing that, if A is a ${\displaystyle n\times n}$ matrix with entries in a field F, then ${\displaystyle \mathbf {A} \mathbf {B} =\mathbf {B} \mathbf {A} }$ for every ${\displaystyle n\times n}$ matrix B with entries in F, if and only if ${\displaystyle \mathbf {A} =c\,\mathbf {I} }$ where ${\displaystyle c\in F}$, and I is the ${\displaystyle n\times n}$ identity matrix. If, instead of a field, the entries are supposed to belong to belong to a ring, then one must add the condition that c belongs to the center of the ring.

### Distributivity

The matrix product is distributive with respect of matrix addition. That is, if A, B, C, D are matrices of respective sizes m × n, n × p, n × p, and p × q, one has (left distributivity)

${\displaystyle \mathbf {A} (\mathbf {B} +\mathbf {C} )=\mathbf {AB} +\mathbf {AC} ,}$

and (right distributivity)

${\displaystyle (\mathbf {B} +\mathbf {C} )\mathbf {D} =\mathbf {BD} +\mathbf {CD} .}$

This results from the distributivity for coefficients by

${\displaystyle \sum _{k}a_{ik}(b_{kj}+c_{kj})=\sum _{k}a_{ik}b_{kj}+\sum _{k}a_{ik}c_{kj}}$
${\displaystyle \sum _{k}(b_{ik}+c_{ik})d_{kj}=\sum _{k}b_{ik}d_{kj}+\sum _{k}c_{ik}d_{kj}.}$

### Product with a scalar

If A is a matrix and c a scalar, then the matrices ${\displaystyle c\mathbf {A} }$ and ${\displaystyle \mathbf {A} c}$ are obtained by left of right multiplying all entries of A by c. If the scalars have the commutative property, then ${\displaystyle c\mathbf {A} =\mathbf {A} c.}$

If the product ${\displaystyle \mathbf {AB} }$ is defined (that is the number of columns of A equals the number of rows of B, then

${\displaystyle c(\mathbf {AB} )=(c\mathbf {A} )\mathbf {B} }$ and ${\displaystyle (\mathbf {A} \mathbf {B} )c=\mathbf {A} (\mathbf {B} c).}$

If the scalars have the commutative property, then all four matrices are equal. More generally, all four are equal if c belongs to the center of a ring containing the entries of the matrices, because in this case cX = Xc for all matrices X.

These properties result from the bilinearity of the product of scalars:

${\displaystyle c\left(\sum _{k}a_{ik}b_{kj}\right)=\sum _{k}(ca_{ik})b_{kj}}$
${\displaystyle \left(\sum _{k}a_{ik}b_{kj}\right)c=\sum _{k}a_{ik}(b_{kj}c).}$

### Transpose

If the scalars have the commutative property, the transpose of a product of matrices is the product, in the reverse order, of the transposes of the factors. That is

${\displaystyle (\mathbf {AB} )^{\mathsf {T}}=\mathbf {B} ^{\mathsf {T}}\mathbf {A} ^{\mathsf {T}}}$

where T denotes the transpose, that is the interchange of rows and columns.

This identity does not hold for noncommutative entries, since the order between the entries of A and B is reversed, when one expands the definition of the matrix product.

### Complex conjugate

If A and B have complex entries, then

${\displaystyle (\mathbf {AB} )^{*}=\mathbf {A} ^{*}\mathbf {B} ^{*}}$

where * denotes the entry-wise complex conjugate of a matrix.

This results of applying to the definition of matrix product the fact that the conjugate of a sum is the sum of the conjugates of the summands and the conjugate of a product is the product of the conjugates of the factors.

Transposition acts on the indices of the entries, while conjugation acts independently on the entries themselves. It results that, if A and B have complex entries, one has

${\displaystyle (\mathbf {AB} )^{\dagger }=\mathbf {B} ^{\dagger }\mathbf {A} ^{\dagger },}$

where denotes the conjugate transpose (conjugate of the transpose, or equivalently transpose of the conjugate).

### Associativity

Given three matrices A, B and C, the products (AB)C and A(BC) are defined if and only the number of columns of A equals the number of rows of B and the number of columns of B equals the number of rows of C (in particular, if one of the product is defined, the other is also defined). In this case, one has the associative property

${\displaystyle (\mathbf {AB} )\mathbf {C} =\mathbf {A} (\mathbf {BC} ).}$

As for any associative operation, this allows omitting parentheses, and writing the above products as ${\displaystyle \mathbf {ABC} .}$

This extends naturally to the product of any number of matrix provided that the dimension match. That is, if A1, A2, ..., An are matrices such that the number of columns of Ai equals the number of rows of Ai + 1 for i = 1, ..., n – 1, then the product

${\displaystyle \prod _{i=1}^{n}\mathbf {A} _{i}=\mathbf {A} _{1}\mathbf {A} _{2}\cdots \mathbf {A} _{n}}$

is defined and does not depend on the order of the multiplications, if the order of the matrices is kept fixed.

These properties may be proved by straightforward but complicate summation manipulations. This result also from the fact that matrices represent linear maps. Therefore, the associative property of matrices is simply a specific case of the associative property of function composition.

#### Complexity is not associative

Although the result of a sequence of matrix product does not depends on the order of operation (provided that the order of the matrices is not changed), the computational complexity may depend dramatically on this order.

For example, if A, B and C are matrices of respective sizes 10×30, 30×5, 5×60, computing (AB)C needs 10×30×5 + 10×5×60 = 4,500 multiplications, while computing A(BC) needs 30×5×60 + 10×30×60 = 27,000 multiplications.

Algorithms have been designed for choosing the best order of products, see Matrix chain multiplication. When the number n of matrices increases, it has been shown that the choice of the best order has a complexity of ${\displaystyle O(n\log n).}$

#### Application to similarity

Any invertible matrix ${\displaystyle \mathbf {P} }$ defines a similarity transformation (on square matrices of the same size as ${\displaystyle \mathbf {P} }$)

${\displaystyle S_{\mathbf {P} }(\mathbf {A} )=\mathbf {P} ^{-1}\mathbf {A} \mathbf {P} .}$

Similarity transformations map product to products, that is

${\displaystyle S_{\mathbf {P} }(\mathbf {AB} )=S_{\mathbf {P} }(\mathbf {B} )S_{\mathbf {P} }(\mathbf {B} ).}$

In fact, one has

${\displaystyle \mathbf {P} ^{-1}(\mathbf {AB} )\mathbf {P} =\mathbf {P} ^{-1}\mathbf {A} (\mathbf {P} \mathbf {P} ^{-1})\mathbf {B} \mathbf {P} =(\mathbf {P} ^{-1}\mathbf {A} \mathbf {P} )(\mathbf {P} ^{-1}\mathbf {B} \mathbf {P} ).}$

## Square matrices

### Identity element

If A is a square matrix, then

${\displaystyle \mathbf {AI} =\mathbf {IA} =\mathbf {A} }$

where I is the identity matrix of the same order.

### Inverse matrix

If A is a square matrix, there may be an inverse matrix A−1 of A such that

${\displaystyle \mathbf {A} \mathbf {A} ^{-1}=\mathbf {A} ^{-1}\mathbf {A} =\mathbf {I} }$

If this property holds then A is an invertible matrix, if not A is a singular matrix. Moreover,

${\displaystyle (\mathbf {AB} )^{\mathrm {-1} }=\mathbf {B} ^{\mathrm {-1} }\mathbf {A} ^{\mathrm {-1} }}$

### Determinants

When a determinant of a matrix is defined (i.e., when the underlying ring is commutative), if A and B are square matrices of the same order, the determinant of their product AB equals the product of their determinants:

${\displaystyle \det(\mathbf {AB} )=\det(\mathbf {A} )\det(\mathbf {B} )}$

Since det(A) and det(B) are elements of the said commutative ring, det(A)det(B) = det(B)det(A), and so det(AB) = det(BA), even when ABBA. }}

### Trace

The trace of a product AB is independent of the order of A and B:

${\displaystyle \mathrm {tr} (\mathbf {AB} )=\mathrm {tr} (\mathbf {BA} )}$

In index notation:

{\displaystyle {\begin{aligned}\mathrm {tr} (\mathbf {AB} )&=\sum _{i}\sum _{k}A_{ik}B_{ki}\\&=\sum _{k}\sum _{i}B_{ki}A_{ik}\\&=\mathrm {tr} (\mathbf {BA} )\end{aligned}}}

## Operations derived from the matrix product

More operations on square matrices can be defined using the matrix product, such as powers and nth roots by repeated matrix products, the matrix exponential can be defined by a power series, the matrix logarithm is the inverse of matrix exponentiation, and so on.

### Powers of matrices

Square matrices can be multiplied by themselves repeatedly in the same way as ordinary numbers, because they always have the same number of rows and columns. This repeated multiplication can be described as a power of the matrix, a special case of the ordinary matrix product. On the contrary, rectangular matrices do not have the same number of rows and columns so they can never be raised to a power. An n × n matrix A raised to a positive integer k is defined as

${\displaystyle \mathbf {A} ^{k}={\underset {k\mathrm {\,times} }{\mathbf {A} \mathbf {A} \cdots \mathbf {A} }}}$

and the following identities hold, where λ is a scalar:

1. Zero power:
${\displaystyle \mathbf {A} ^{0}=\mathbf {I} }$
where I is the identity matrix. This is parallel to the zeroth power of any number which equals unity.
2. Scalar multiplication:
${\displaystyle (\lambda \mathbf {A} )^{k}=\lambda ^{k}\mathbf {A} ^{k}}$
3. Determinant:
${\displaystyle \det(\mathbf {A} ^{k})=\det(\mathbf {A} )^{k}}$

The naive computation of matrix powers is to multiply k times the matrix A to the result, starting with the identity matrix just like the scalar case. This can be improved using exponentiation by squaring, a method commonly used for scalars. For diagonalizable matrices, an even better method is to use the eigenvalue decomposition of A. Another method based on the Cayley–Hamilton theorem finds an identity using the matrices' characteristic polynomial, producing a more effective equation for Ak in which a scalar is raised to the required power, rather than an entire matrix.

A special case is the power of a diagonal matrix. Since the product of diagonal matrices amounts to simply multiplying corresponding diagonal elements together, the power k of a diagonal matrix A will have entries raised to the power. Explicitly;

${\displaystyle \mathbf {A} ^{k}={\begin{pmatrix}A_{11}&0&\cdots &0\\0&A_{22}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &A_{nn}\end{pmatrix}}^{k}={\begin{pmatrix}A_{11}^{k}&0&\cdots &0\\0&A_{22}^{k}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &A_{nn}^{k}\end{pmatrix}}}$

meaning it is easy to raise a diagonal matrix to a power. When raising an arbitrary matrix (not necessarily a diagonal matrix) to a power, it is often helpful to exploit this property by diagonalizing the matrix first.

## Applications of the matrix product

### Linear transformations

Matrices offer a concise way of representing linear transformations between vector spaces, and matrix multiplication corresponds to the composition of linear transformations. The matrix product of two matrices can be defined when their entries belong to the same ring, and hence can be added and multiplied.

Let U, V, and W be vector spaces over the same field with given bases, S: VW and T: UV be linear transformations and ST: UW be their composition.

Suppose that A, B, and C are the matrices representing the transformations S, T, and ST with respect to the given bases.

Then AB = C, that is, the matrix of the composition (or the product) of linear transformations is the product of their matrices with respect to the given bases.

### Linear systems of equations

A system of linear equations with the same number of equations as variables can be solved by collecting the coefficients of the equations into a square matrix, then inverting the matrix equation.

A similar procedure can be used to solve a system of linear differential equations, see also phase plane.

## The inner and outer products

Given two column vectors a and b, the Euclidean inner product and outer product are the simplest special cases of the matrix product.[9]

### Inner product

The inner product of two vectors in matrix form is equivalent to a column vector multiplied on its left by a row vector:

{\displaystyle {\begin{aligned}\mathbf {a} \cdot \mathbf {b} &=\mathbf {a} ^{\mathrm {T} }\mathbf {b} \\&={\begin{pmatrix}a_{1}&a_{2}&\cdots &a_{n}\end{pmatrix}}{\begin{pmatrix}b_{1}\\b_{2}\\\vdots \\b_{n}\end{pmatrix}}\\&=a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}\\&=\sum _{i=1}^{n}a_{i}b_{i},\end{aligned}}}

where aT denotes the transpose of a.

The matrix product itself can be expressed in terms of inner product. Suppose that the first n × m matrix A is decomposed into its row vectors ai, and the second m × p matrix B into its column vectors bi:[1]

${\displaystyle \mathbf {A} ={\begin{pmatrix}A_{11}&A_{12}&\cdots &A_{1m}\\A_{21}&A_{22}&\cdots &A_{2m}\\\vdots &\vdots &\ddots &\vdots \\A_{n1}&A_{n2}&\cdots &A_{nm}\end{pmatrix}}={\begin{pmatrix}\mathbf {a} _{1}\\\mathbf {a} _{2}\\\vdots \\\mathbf {a} _{n}\end{pmatrix}},}$
${\displaystyle \mathbf {B} ={\begin{pmatrix}B_{11}&B_{12}&\cdots &B_{1p}\\B_{21}&B_{22}&\cdots &B_{2p}\\\vdots &\vdots &\ddots &\vdots \\B_{m1}&B_{m2}&\cdots &B_{mp}\end{pmatrix}}={\begin{pmatrix}\mathbf {b} _{1}&\mathbf {b} _{2}&\cdots &\mathbf {b} _{p}\end{pmatrix}}}$

where

${\displaystyle \mathbf {a} _{i}={\begin{pmatrix}A_{i1}&A_{i2}&\cdots &A_{im}\end{pmatrix}}\,,\quad \mathbf {b} _{i}={\begin{pmatrix}B_{1i}\\B_{2i}\\\vdots \\B_{mi}\end{pmatrix}}}$

Then:

${\displaystyle \mathbf {AB} ={\begin{pmatrix}\mathbf {a} _{1}\\\mathbf {a} _{2}\\\vdots \\\mathbf {a} _{n}\end{pmatrix}}{\begin{pmatrix}\mathbf {b} _{1}&\mathbf {b} _{2}&\dots &\mathbf {b} _{p}\end{pmatrix}}={\begin{pmatrix}(\mathbf {a} _{1}\cdot \mathbf {b} _{1})&(\mathbf {a} _{1}\cdot \mathbf {b} _{2})&\dots &(\mathbf {a} _{1}\cdot \mathbf {b} _{p})\\(\mathbf {a} _{2}\cdot \mathbf {b} _{1})&(\mathbf {a} _{2}\cdot \mathbf {b} _{2})&\dots &(\mathbf {a} _{2}\cdot \mathbf {b} _{p})\\\vdots &\vdots &\ddots &\vdots \\(\mathbf {a} _{n}\cdot \mathbf {b} _{1})&(\mathbf {a} _{n}\cdot \mathbf {b} _{2})&\dots &(\mathbf {a} _{n}\cdot \mathbf {b} _{p})\end{pmatrix}}}$

It is also possible to express a matrix product in terms of concatenations of products of matrices and row or column vectors:

${\displaystyle \mathbf {AB} ={\begin{pmatrix}\mathbf {A} \mathbf {b} _{1}&\mathbf {A} \mathbf {b} _{2}&\dots &\mathbf {A} \mathbf {b} _{p}\end{pmatrix}}={\begin{pmatrix}\mathbf {a} _{1}\mathbf {B} \\\mathbf {a} _{2}\mathbf {B} \\\vdots \\\mathbf {a} _{n}\mathbf {B} \end{pmatrix}}}$

These decompositions are particularly useful for matrices that are envisioned as concatenations of particular types of row vectors or column vectors, e.g. orthogonal matrices (whose rows and columns are unit vectors orthogonal to each other) and Markov matrices (whose rows or columns sum to 1).[citation needed]

### Outer product

The outer product (also known as the dyadic product or tensor product) of two vectors in matrix form is equivalent to a row vector multiplied on the left by a column vector:

{\displaystyle {\begin{aligned}\mathbf {a} \otimes \mathbf {b} &=\mathbf {a} \mathbf {b} ^{\mathrm {T} }\\&={\begin{pmatrix}a_{1}\\a_{2}\\\vdots \\a_{n}\end{pmatrix}}{\begin{pmatrix}b_{1}&b_{2}&\cdots &b_{n}\end{pmatrix}}\\&={\begin{pmatrix}a_{1}b_{1}&a_{1}b_{2}&\cdots &a_{1}b_{n}\\a_{2}b_{1}&a_{2}b_{2}&\cdots &a_{2}b_{n}\\\vdots &\vdots &\ddots &\vdots \\a_{n}b_{1}&a_{n}b_{2}&\cdots &a_{n}b_{n}\\\end{pmatrix}}.\end{aligned}}}

An alternative method is to express the matrix product in terms of the outer product. The decomposition is done the other way around, the first matrix A is decomposed into column vectors ai and the second matrix B into row vectors bi:

{\displaystyle {\begin{aligned}\mathbf {AB} &={\begin{pmatrix}\mathbf {\bar {a}} _{1}&\mathbf {\bar {a}} _{2}&\cdots &\mathbf {\bar {a}} _{m}\end{pmatrix}}{\begin{pmatrix}\mathbf {\bar {b}} _{1}\\\mathbf {\bar {b}} _{2}\\\vdots \\\mathbf {\bar {b}} _{m}\end{pmatrix}}\\&=\mathbf {\bar {a}} _{1}\otimes \mathbf {\bar {b}} _{1}+\mathbf {\bar {a}} _{2}\otimes \mathbf {\bar {b}} _{2}+\cdots +\mathbf {\bar {a}} _{m}\otimes \mathbf {\bar {b}} _{m}\\&=\sum _{i=1}^{m}\mathbf {\bar {a}} _{i}\otimes \mathbf {\bar {b}} _{i}\end{aligned}}}

where this time

${\displaystyle \mathbf {\bar {a}} _{i}={\begin{pmatrix}A_{1i}\\A_{2i}\\\vdots \\A_{ni}\end{pmatrix}}\,,\quad \mathbf {\bar {b}} _{i}={\begin{pmatrix}B_{i1}&B_{i2}&\cdots &B_{ip}\end{pmatrix}}\,.}$

This method emphasizes the effect of individual column/row pairs on the result, which is a useful point of view with e.g. covariance matrices, where each such pair corresponds to the effect of a single sample point.[citation needed]

{\displaystyle {\begin{aligned}{\begin{pmatrix}{\color {Brown}1}&{\color {Orange}2}&{\color {Violet}3}\\{\color {Brown}4}&{\color {Orange}5}&{\color {Violet}6}\\{\color {Brown}7}&{\color {Orange}8}&{\color {Violet}9}\\\end{pmatrix}}{\begin{pmatrix}{\color {Brown}a}&{\color {Brown}d}\\{\color {Orange}b}&{\color {Orange}e}\\{\color {Violet}c}&{\color {Violet}f}\\\end{pmatrix}}&={\begin{pmatrix}{\color {Brown}1}\\{\color {Brown}4}\\{\color {Brown}7}\\\end{pmatrix}}\otimes {\begin{pmatrix}{\color {Brown}{a}}&{\color {Brown}{d}}\\\end{pmatrix}}+{\begin{pmatrix}{\color {Orange}2}\\{\color {Orange}5}\\{\color {Orange}8}\\\end{pmatrix}}\otimes {\begin{pmatrix}{\color {Orange}{b}}&{\color {Orange}{e}}\\\end{pmatrix}}+{\begin{pmatrix}{\color {Violet}3}\\{\color {Violet}6}\\{\color {Violet}9}\\\end{pmatrix}}\otimes {\begin{pmatrix}{\color {Violet}c}&{\color {Violet}f}\\\end{pmatrix}}\\&={\begin{pmatrix}{\color {Brown}{1a}}&{\color {Brown}{1d}}\\{\color {Brown}{4a}}&{\color {Brown}{4d}}\\{\color {Brown}{7a}}&{\color {Brown}{7d}}\\\end{pmatrix}}+{\begin{pmatrix}{\color {Orange}{2b}}&{\color {Orange}{2e}}\\{\color {Orange}{5b}}&{\color {Orange}{5e}}\\{\color {Orange}{8b}}&{\color {Orange}{8e}}\\\end{pmatrix}}+{\begin{pmatrix}{\color {Violet}{3c}}&{\color {Violet}{3f}}\\{\color {Violet}{6c}}&{\color {Violet}{6f}}\\{\color {Violet}{9c}}&{\color {Violet}{9f}}\\\end{pmatrix}}\\&={\begin{pmatrix}{\color {Brown}{1a}}+{\color {Orange}{2b}}+{\color {Violet}{3c}}&{\color {Brown}{1d}}+{\color {Orange}{2e}}+{\color {Violet}{3f}}\\{\color {Brown}{4a}}+{\color {Orange}{5b}}+{\color {Violet}{6c}}&{\color {Brown}{4d}}+{\color {Orange}{5e}}+{\color {Violet}{6f}}\\{\color {Brown}{7a}}+{\color {Orange}{8b}}+{\color {Violet}{9c}}&{\color {Brown}{7d}}+{\color {Orange}{8e}}+{\color {Violet}{9f}}\\\end{pmatrix}}.\end{aligned}}}

## Algorithms for efficient matrix multiplication

The bound on ω over time.

The running time of square matrix multiplication, if carried out naïvely, is O(n3). The running time for multiplying rectangular matrices (one m × p-matrix with one p × n-matrix) is O(mnp), however, more efficient algorithms exist, such as Strassen's algorithm, devised by Volker Strassen in 1969 and often referred to as "fast matrix multiplication". It is based on a way of multiplying two 2 × 2-matrices which requires only 7 multiplications (instead of the usual 8), at the expense of several additional addition and subtraction operations. Applying this recursively gives an algorithm with a multiplicative cost of ${\displaystyle O(n^{\log _{2}7})\approx O(n^{2.807})}$. Strassen's algorithm is more complex, and the numerical stability is reduced compared to the naïve algorithm.[10] Nevertheless, it appears in several libraries, such as BLAS, where it is significantly more efficient for matrices with dimensions n > 100,[11] and is very useful for large matrices over exact domains such as finite fields, where numerical stability is not an issue.

The current O(nk) algorithm with the lowest known exponent k is a generalization of the Coppersmith–Winograd algorithm that has an asymptotic complexity of O(n2.3728639), by François Le Gall.[12] This algorithm, and the Coppersmith–Winograd algorithm on which it is based, are similar to Strassen's algorithm: a way is devised for multiplying two k × k-matrices with fewer than k3 multiplications, and this technique is applied recursively. However, the constant coefficient hidden by the Big O notation is so large that these algorithms are only worthwhile for matrices that are too large to handle on present-day computers.[13][14]

Since any algorithm for multiplying two n × n-matrices has to process all 2 × n2-entries, there is an asymptotic lower bound of Ω(n2) operations. Raz (2002) proves a lower bound of Ω(n2 log(n)) for bounded coefficient arithmetic circuits over the real or complex numbers.

Cohn et al. (2003, 2005) put methods such as the Strassen and Coppersmith–Winograd algorithms in an entirely different group-theoretic context, by utilising triples of subsets of finite groups which satisfy a disjointness property called the triple product property (TPP). They show that if families of wreath products of Abelian groups with symmetric groups realise families of subset triples with a simultaneous version of the TPP, then there are matrix multiplication algorithms with essentially quadratic complexity. Most researchers initially believed that this was indeed the case.[15] However, Blasiak, Church, Cohn, Grochow, Naslund, Sawin and Umans have recently shown using the Slice Rank method that such an approach using wreath products of abelian groups with small exponent, cannot yield a matrix multiplication exponent of 2. [16]

Freivalds' algorithm is a simple Monte Carlo algorithm that given matrices A, B, C verifies in Θ(n2) time if AB = C.

Block matrix multiplication. In the 2D algorithm, each processor is responsible for one submatrix of C. In the 3D algorithm, every pair of submatrices from A and B that is multiplied is assigned to one processor.

### Parallel matrix multiplication

Because of the nature of matrix operations and the layout of matrices in memory, it is typically possible to gain substantial performance gains through use of parallelization and vectorization. Several algorithms are possible, among which divide and conquer algorithms based on the block matrix decomposition

${\displaystyle \mathbf {C} ={\begin{pmatrix}\mathbf {C} _{11}&\mathbf {C} _{12}\\\mathbf {C} _{21}&\mathbf {C} _{22}\\\end{pmatrix}}={\begin{pmatrix}\mathbf {A} _{11}&\mathbf {A} _{12}\\\mathbf {A} _{21}&\mathbf {A} _{22}\\\end{pmatrix}}{\begin{pmatrix}\mathbf {B} _{11}&\mathbf {B} _{12}\\\mathbf {B} _{21}&\mathbf {B} _{22}\\\end{pmatrix}}=\mathbf {A} \mathbf {B} }$

that also underlies Strassen's algorithm. Here, A, B and C are presumed to be n by n (square) matrices, and C11 etc. are n/2 by n/2 submatrices. From this decomposition, one derives[17]

${\displaystyle {\begin{pmatrix}\mathbf {A} _{11}&\mathbf {A} _{12}\\\mathbf {A} _{21}&\mathbf {A} _{22}\\\end{pmatrix}}{\begin{pmatrix}\mathbf {B} _{11}&\mathbf {B} _{12}\\\mathbf {B} _{21}&\mathbf {B} _{22}\\\end{pmatrix}}={\begin{pmatrix}\mathbf {A} _{11}\mathbf {B} _{11}+\mathbf {A} _{12}\mathbf {B} _{21}&\mathbf {A} _{11}\mathbf {B} _{12}+\mathbf {A} _{12}\mathbf {B} _{22}\\\mathbf {A} _{21}\mathbf {B} _{11}+\mathbf {A} _{22}\mathbf {B} _{21}&\mathbf {A} _{21}\mathbf {B} _{12}+\mathbf {A} _{22}\mathbf {B} _{22}\\\end{pmatrix}}}$

which consists of eight multiplications of pairs of submatrices, which can all be performed in parallel, followed by an addition step. Applying this recursively, and performing the additions in parallel as well, one obtains an algorithm that runs in Θ(log2 n) time on an ideal machine with an infinite number of processors, and has a maximum possible speedup of Θ(n3/(log2 n)) on any real computer (although the algorithm isn't practical, a more practical variant achieves Θ(n2) speedup).[17]

It should be noted that some lower time-complexity algorithms on paper may have indirect time complexity costs on real machines.

### Communication-avoiding and distributed algorithms

On modern architectures with hierarchical memory, the cost of loading and storing input matrix elements tends to dominate the cost of arithmetic. On a single machine this is the amount of data transferred between RAM and cache, while on a distributed memory multi-node machine it is the amount transferred between nodes; in either case it is called the communication bandwidth. The naïve algorithm using three nested loops uses Ω(n3) communication bandwidth.

Cannon's algorithm, also known as the 2D algorithm, partitions each input matrix into a block matrix whose elements are submatrices of size M/3 by M/3, where M is the size of fast memory.[18] The naïve algorithm is then used over the block matrices, computing products of submatrices entirely in fast memory. This reduces communication bandwidth to O(n3/M), which is asymptotically optimal (for algorithms performing Ω(n3) computation).[19][20]

In a distributed setting with p processors arranged in a p by p 2D mesh, one submatrix of the result can be assigned to each processor, and the product can be computed with each processor transmitting O(n2/p) words, which is asymptotically optimal assuming that each node stores the minimum O(n2/p) elements.[20] This can be improved by the 3D algorithm, which arranges the processors in a 3D cube mesh, assigning every product of two input submatrices to a single processor. The result submatrices are then generated by performing a reduction over each row.[21] This algorithm transmits O(n2/p2/3) words per processor, which is asymptotically optimal.[20] However, this requires replicating each input matrix element p1/3 times, and so requires a factor of p1/3 more memory than is needed to store the inputs. This algorithm can be combined with Strassen to further reduce runtime.[21] "2.5D" algorithms provide a continuous tradeoff between memory usage and communication bandwidth.[22] On modern distributed computing environments such as MapReduce, specialized multiplication algorithms have been developed.[23]

## Other forms of multiplication

The term "matrix multiplication" is most commonly reserved for the definition given in this article. It could be more loosely applied to other definitions.

## Notes

1. ^ a b Lerner, R. G.; Trigg, G. L. (1991). Encyclopaedia of Physics (2nd ed.). VHC publishers. ISBN 3-527-26954-1.
2. ^ Parker, C. B. (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). ISBN 0-07-051400-3.
3. ^ Lipschutz, S.; Lipson, M. (2009). Linear Algebra. Schaum's Outlines (4th ed.). McGraw Hill (USA). pp. 30–31. ISBN 978-0-07-154352-1.
4. ^ Riley, K. F.; Hobson, M. P.; Bence, S. J. (2010). Mathematical methods for physics and engineering. Cambridge University Press. ISBN 978-0-521-86153-3.
5. ^ Adams, R. A. (1995). Calculus, A Complete Course (3rd ed.). Addison Wesley. p. 627. ISBN 0 201 82823 5.
6. ^ Horn, Johnson (2013). Matrix Analysis (2nd ed.). Cambridge University Press. p. 6. ISBN 978 0 521 54823 6.
7. ^ Lipcshutz, S.; Lipson, M. (2009). "2". Linear Algebra. Schaum's Outlines (4th ed.). McGraw Hill (USA). ISBN 978-0-07-154352-1.
8. ^ Horn, Johnson (2013). "0". Matrix Analysis (2nd ed.). Cambridge University Press. ISBN 978 0 521 54823 6.
9. ^ Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISBN 978-0-521-86153-3
10. ^ Miller, Webb (1975), "Computational complexity and numerical stability", SIAM News, 4: 97–107, CiteSeerX , doi:10.1137/0204009
11. ^ Press 2007, p. 108.
12. ^ Le Gall, François (2014), "Powers of tensors and fast matrix multiplication", Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation (ISSAC 2014), arXiv:. The original algorithm was presented by Don Coppersmith and Shmuel Winograd in 1990, has an asymptotic complexity of O(n2.376). It was improved in 2013 to O(n2.3729) by Virginia Vassilevska Williams, giving a time only slightly worse than Le Gall's improvement: Williams, Virginia Vassilevska. "Multiplying matrices faster than Coppersmith-Winograd" (PDF).
13. ^ Iliopoulos, Costas S. (1989), "Worst-case complexity bounds on algorithms for computing the canonical structure of finite abelian groups and the Hermite and Smith normal forms of an integer matrix" (PDF), SIAM Journal on Computing, 18 (4): 658–669, doi:10.1137/0218045, MR 1004789, The Coppersmith–Winograd algorithm is not practical, due to the very large hidden constant in the upper bound on the number of multiplications required.
14. ^ Robinson, Sara (2005), "Toward an Optimal Algorithm for Matrix Multiplication" (PDF), SIAM News, 38 (9)
15. ^ Robinson, 2005.
16. ^ [Jonah Blasiak, Thomas Church, Henry Cohn, Joshua A. Grochow, Eric Naslund, William F. Sawin, Chris Umans, On cap sets and the group-theoretic approach to matrix multiplication
17. ^ a b Randall, Keith H. (1998). Cilk: Efficient Multithreaded Computing (PDF) (Ph.D.). Massachusetts Institute of Technology. pp. 54–57.
18. ^ Lynn Elliot Cannon, A cellular computer to implement the Kalman Filter Algorithm, Technical report, Ph.D. Thesis, Montana State University, 14 July 1969.
19. ^ Hong, J.W.; Kung, H. T. (1981). "I/O complexity: The red-blue pebble game". STOC ’81: Proceedings of the thirteenth annual ACM symposium on Theory of computing: 326–333.
20. ^ a b c Irony, Dror; Toledo, Sivan; Tiskin, Alexander (September 2004). "Communication lower bounds for distributed-memory matrix multiplication". J. Parallel Distrib. Comput. 64 (9): 1017–1026. doi:10.1016/j.jpdc.2004.03.021.
21. ^ a b Agarwal, R.C.; Balle, S. M.; Gustavson, F. G.; Joshi, M.; Palkar, P. (September 1995). "A three-dimensional approach to parallel matrix multiplication". IBM J. Res. Dev. 39 (5): 575–582. doi:10.1147/rd.395.0575.
22. ^ Solomonik, Edgar; Demmel, James (2011). "Communication-optimal parallel 2.5D matrix multiplication and LU factorization algorithms". Proceedings of the 17th international conference on Parallel processing. Part II: 90–109. doi:10.1007/978-3-642-23397-5_10.
23. ^ Pietracaprina, A.; Pucci, G.; Riondato, M.; Silvestri, F.; Upfal, E. (2012). "Space-Round Tradeoffs for MapReduce Computations". Proc. of 26th ACM International Conference on Supercomputing. Venice (Italy): ACM. pp. 235–244. doi:10.1145/2304576.2304607.

## References

• Henry Cohn, Robert Kleinberg, Balázs Szegedy, and Chris Umans. Group-theoretic Algorithms for Matrix Multiplication. arXiv:math.GR/0511460. Proceedings of the 46th Annual Symposium on Foundations of Computer Science, 23–25 October 2005, Pittsburgh, PA, IEEE Computer Society, pp. 379–388.
• Henry Cohn, Chris Umans. A Group-theoretic Approach to Fast Matrix Multiplication. arXiv:math.GR/0307321. Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, 11–14 October 2003, Cambridge, MA, IEEE Computer Society, pp. 438–449.
• Coppersmith, D.; Winograd, S. (1990). "Matrix multiplication via arithmetic progressions". J. Symbolic Comput. 9: 251–280. doi:10.1016/s0747-7171(08)80013-2.
• Horn, Roger A.; Johnson, Charles R. (1991), Topics in Matrix Analysis, Cambridge University Press, ISBN 978-0-521-46713-1
• Knuth, D.E., The Art of Computer Programming Volume 2: Seminumerical Algorithms. Addison-Wesley Professional; 3 edition (November 14, 1997). ISBN 978-0-201-89684-8. pp. 501.
• Press, William H.; Flannery, Brian P.; Teukolsky, Saul A.; Vetterling, William T. (2007), Numerical Recipes: The Art of Scientific Computing (3rd ed.), Cambridge University Press, ISBN 978-0-521-88068-8.
• Ran Raz. On the complexity of matrix product. In Proceedings of the thirty-fourth annual ACM symposium on Theory of computing. ACM Press, 2002. doi:10.1145/509907.509932.
• Robinson, Sara, Toward an Optimal Algorithm for Matrix Multiplication, SIAM News 38(9), November 2005. PDF
• Strassen, Volker, Gaussian Elimination is not Optimal, Numer. Math. 13, p. 354-356, 1969.
• Styan, George P. H. (1973), "Hadamard Products and Multivariate Statistical Analysis", Linear Algebra and its Applications, 6: 217–240, doi:10.1016/0024-3795(73)90023-2
• Vassilevska Williams, Virginia, Multiplying matrices faster than Coppersmith-Winograd, Manuscript, May 2012. PDF