Matrix multiplication#Matrix product .28two matrices.29

This article will use the following notational conventions: matrices are represented by capital letters in bold, e.g. {{math|A}}; vectors in lowercase bold, e.g. {{math|a}}; and entries of vectors and matrices are italic (they are numbers from a field), e.g. {{math|A}} and {{math|a}}. Index notation is often the clearest way to express definitions, and is used as standard in the literature. The entry in row {{mvar|i}}, column {{mvar|j}} of matrix {{math|A}} is indicated by {{math|(A)_ij}}, {{math|A_ij}} or {{math|a_ij}}. In contrast, a single subscript, e.g. {{math|A₁, A₂}}, is used to select a matrix (not a matrix entry) from a collection of matrices.

If {{math|A}} is an {{math|m × n}} matrix and {{math|B}} is an {{math|n × p}} matrix,

$$\backslash mathbf\{A\}=\backslash begin\{pmatrix\}$$

a_{11} & a_{12} & \cdots & a_{1n} \\

a_{21} & a_{22} & \cdots & a_{2n} \\

\vdots & \vdots & \ddots & \vdots \\

a_{m1} & a_{m2} & \cdots & a_{mn} \\

\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_{n1} & b_{n2} & \cdots & b_{np} \\

\end{pmatrix}

the matrix product {{math|1=C = AB}} (denoted without multiplication signs or dots) is defined to be the {{math|m × p}} matrix{{cite book| title=Linear Algebra | edition=4th | first1 = S. | last1 = Lipschutz | first2 = M. | last2 = Lipson|series=Schaum's Outlines | publisher=McGraw Hill (USA) | date=2009 | pages=30–31 | isbn=978-0-07-154352-1}}{{cite book | title=Mathematical methods for physics and engineering | url=https://archive.org/details/mathematicalmeth00rile | url-access = registration | first1 = K. F. | last1 = Riley | first2 = M. P. | last2 = Hobson | first3 = S. J. | last3 = Bence| publisher=Cambridge University Press | date=2010 | isbn=978-0-521-86153-3}}{{cite book | title=Calculus, A Complete Course | edition=3rd| first = R. A. | last = Adams|publisher=Addison Wesley |date=1995 |page=627 |isbn=0-201-82823-5}}{{cite book|title=Matrix Analysis | last = Horn | first = Johnson |edition=2nd | publisher=Cambridge University Press | date=2013 |page=6 |isbn=978-0-521-54823-6}}

$$\backslash mathbf\{C\}\; =\; \backslash begin\{pmatrix\}$$

c_{11} & c_{12} & \cdots & c_{1p} \\

c_{21} & c_{22} & \cdots & c_{2p} \\

\vdots & \vdots & \ddots & \vdots \\

c_{m1} & c_{m2} & \cdots & c_{mp} \\

\end{pmatrix}

such that

$$c\_\{ij\}\; =\; a\_\{i1\}\; b\_\{1j\}\; +\; a\_\{i2\}\; b\_\{2j\}\; +\; \backslash cdots\; +\; a\_\{in\}\; b\_\{nj\}\; =\; \backslash sum\_\{k=1\}^n\; a\_\{ik\}\; b\_\{kj\},$$

for {{math|1=i = 1, ..., m}} and {{math|1=j = 1, ..., p}}.

That is, the entry {{tmath|c_{ij} }} of the product is obtained by multiplying term-by-term the entries of the {{mvar|i}}th row of {{math|A}} and the {{mvar|j}}th column of {{math|B}}, and summing these {{mvar|n}} products. In other words, {{tmath|c_{ij} }} is the dot product of the {{mvar|i}}th row of {{math|A}} and the {{mvar|j}}th column of {{math|B}}.

Therefore, {{math|AB}} can also be written as

$$\backslash mathbf\{C\}\; =\; \backslash begin\{pmatrix\}$$

a_{11}b_{11} +\cdots + a_{1n}b_{n1} & a_{11}b_{12} +\cdots + a_{1n}b_{n2} & \cdots & a_{11}b_{1p} +\cdots + a_{1n}b_{np} \\

a_{21}b_{11} +\cdots + a_{2n}b_{n1} & a_{21}b_{12} +\cdots + a_{2n}b_{n2} & \cdots & a_{21}b_{1p} +\cdots + a_{2n}b_{np} \\

\vdots & \vdots & \ddots & \vdots \\

a_{m1}b_{11} +\cdots + a_{mn}b_{n1} & a_{m1}b_{12} +\cdots + a_{mn}b_{n2} & \cdots & a_{m1}b_{1p} +\cdots + a_{mn}b_{np} \\

\end{pmatrix}

Thus the product {{math|AB}} is defined if and only if the number of columns in {{math|A}} equals the number of rows in {{math|B}}, in this case {{math|n}}.

In most scenarios, the entries are numbers, but they may be any kind of 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).

A vector $\backslash mathbf\; x$ of length $n$ can be viewed as a column vector, corresponding to an $n\backslash times1$ matrix $\backslash mathbf\; X$ whose entries are given by $\backslash mathbf\; X\_\{i1\}=\backslash mathbf\; x\_i.$ If $\backslash mathbf\; A$ is an $m\backslash times\; n$ matrix, the matrix-times-vector product denoted by $\backslash mathbf\; \{Ax\}$ is then the vector $\backslash mathbf\; y$ that, viewed as a column vector, is equal to the $m\backslash times1$ matrix $\backslash mathbf\{AX\}.$ In index notation, this amounts to:

:$y\_i=\backslash sum\_\{j=1\}^n\; a\_\{ij\}x\_j.$

One way of looking at this is that the changes from "plain" vector to column vector and back are assumed and left implicit.

Similarly, a vector $\backslash mathbf\; x$ of length $n$ can be viewed as a row vector, corresponding to a $1\backslash times\; n$ matrix. To make it clear that a row vector is meant, it is customary in this context to represent it as the transpose of a column vector; thus, one will see notations such as $\backslash mathbf\{x\}^\backslash mathrm\{T\}\backslash mathbf\{A\}.$ The identity $\backslash mathbf\{x\}^\backslash mathrm\{T\}\backslash mathbf\{A\}=(\backslash mathbf\{A\}^\backslash mathrm\{T\}\backslash mathbf\{x\})^\backslash mathrm\{T\}$ holds. In index notation, if $\backslash mathbf\; A$ is an $n\backslash times\; p$ matrix, $\backslash mathbf\{x\}^\backslash mathrm\{T\}\backslash mathbf\{A\}=\backslash mathbf\{y\}^\backslash mathrm\{T\}$ amounts to:

$y\_k=\backslash sum\_\{j=1\}^n\; x\_j\; a\_\{jk\}.$

The dot product $\backslash mathbf\; a\backslash cdot\backslash mathbf\; b$ of two vectors $\backslash mathbf\; a$ and $\backslash mathbf\; b$ of equal length is equal to the single entry of the $1\backslash times\; 1$ matrix resulting from multiplying these vectors as a row and a column vector, thus: $\backslash mathbf\{a\}^\backslash mathrm\{T\}\backslash mathbf\{b\}$ (or $\backslash mathbf\{b\}^\backslash mathrm\{T\}\backslash mathbf\{a\},$ which results in the same $1\backslash times\; 1$ matrix).

File:Matrix multiplication diagram 2.svg

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

\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 & c_{12} & \cdot \\

\cdot & \cdot & \cdot \\

\cdot & \cdot & c_{33} \\

\cdot & \cdot & \cdot \\

\end{bmatrix}}

The values at the intersections, marked with circles in figure to the right, are:

$$\backslash begin\{align\}$$

c_{12} & = a_{11} b_{12} + a_{12} b_{22} \\

c_{33} & = a_{31} b_{13} + a_{32} b_{23} .

\end{align}

Historically, matrix multiplication has been introduced for facilitating and clarifying computations in linear algebra. This strong relationship between matrix multiplication and linear algebra remains fundamental in all mathematics, as well as in physics, chemistry, engineering and computer science.

If a vector space has a finite basis, its vectors are each uniquely represented by a finite sequence of scalars, called a coordinate vector, whose elements are the coordinates of the vector on the basis. These coordinate vectors form another vector space, which is isomorphic to the original vector space. A coordinate vector is commonly organized as a column matrix (also called a column vector), which is a matrix with only one column. So, a column vector represents both a coordinate vector, and a vector of the original vector space.

A linear map {{mvar|A}} from a vector space of dimension {{mvar|n}} into a vector space of dimension {{mvar|m}} maps a column vector

:$\backslash mathbf\; x=\backslash begin\{pmatrix\}x\_1\; \backslash \backslash \; x\_2\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; x\_n\backslash end\{pmatrix\}$

onto the column vector

:$\backslash mathbf\; y=\; A(\backslash mathbf\; x)=\; \backslash begin\{pmatrix\}a\_\{11\}x\_1+\backslash cdots\; +\; a\_\{1n\}x\_n\backslash \backslash \; a\_\{21\}x\_1+\backslash cdots\; +\; a\_\{2n\}x\_n\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; a\_\{m1\}x\_1+\backslash cdots\; +\; a\_\{mn\}x\_n\backslash end\{pmatrix\}.$

The linear map {{mvar|A}} is thus defined by the matrix

:$\backslash mathbf\{A\}=\backslash begin\{pmatrix\}$

a_{11} & a_{12} & \cdots & a_{1n} \\

a_{21} & a_{22} & \cdots & a_{2n} \\

\vdots & \vdots & \ddots & \vdots \\

a_{m1} & a_{m2} & \cdots & a_{mn} \\

\end{pmatrix},

and maps the column vector $\backslash mathbf\; x$ to the matrix product

:$\backslash mathbf\; y\; =\; \backslash mathbf\; \{Ax\}.$

If {{mvar|B}} is another linear map from the preceding vector space of dimension {{mvar|m}}, into a vector space of dimension {{mvar|p}}, it is represented by a {{tmath|p\times m}} matrix $\backslash mathbf\; B.$ A straightforward computation shows that the matrix of the composite map {{tmath|B\circ A}} is the matrix product $\backslash mathbf\; \{BA\}.$ The general formula {{tmath|1=(B\circ A)(\mathbf x) = B(A(\mathbf x))}}) that defines the function composition is instanced here as a specific case of associativity of matrix product (see {{slink||Associativity}} below):

:$(\backslash mathbf\{BA\})\backslash mathbf\; x\; =\; \backslash mathbf\{B\}(\backslash mathbf\; \{Ax\})\; =\; \backslash mathbf\{BAx\}.$

{{See also|Rotation matrix}}

Using a Cartesian coordinate system in a Euclidean plane, the rotation by an angle $\backslash alpha$ around the origin is a linear map.

More precisely,

$$\backslash begin\{bmatrix\}\; x\text{'}\; \backslash \backslash \; y\text{'}\; \backslash end\{bmatrix\}\; =$$

\begin{bmatrix} \cos \alpha & - \sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix},

where the source point $(x,y)$ and its image $(x\text{'},y\text{'})$ are written as column vectors.

The composition of the rotation by $\backslash alpha$ and that by $\backslash beta$ then corresponds to the matrix product

\begin{bmatrix} \cos \alpha & - \sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix}

= \begin{bmatrix} \cos \beta \cos \alpha - \sin \beta \sin \alpha & - \cos \beta \sin \alpha - \sin \beta \cos \alpha \\

\sin \beta \cos \alpha + \cos \beta \sin \alpha & - \sin \beta \sin \alpha + \cos \beta \cos \alpha \end{bmatrix}

= \begin{bmatrix} \cos (\alpha+\beta) & - \sin(\alpha+\beta) \\ \sin(\alpha+\beta) & \cos(\alpha+\beta) \end{bmatrix},

where appropriate trigonometric identities are employed for the second equality.

That is, the composition corresponds to the rotation by angle $\backslash alpha+\backslash beta$, as expected.

File:Mmult factory svg.svg

As an example, a fictitious factory uses 4 kinds of basic commodities, $b\_1,\; b\_2,\; b\_3,\; b\_4$ to produce 3 kinds of intermediate goods, $m\_1,\; m\_2,\; m\_3$, which in turn are used to produce 3 kinds of final products, $f\_1,\; f\_2,\; f\_3$. The matrices

:   and  

provide the amount of basic commodities needed for a given amount of intermediate goods, and the amount of intermediate goods needed for a given amount of final products, respectively.

For example, to produce one unit of intermediate good $m\_1$, one unit of basic commodity $b\_1$, two units of $b\_2$, no units of $b\_3$, and one unit of $b\_4$ are needed, corresponding to the first column of $\backslash mathbf\{A\}$.

Using matrix multiplication, compute

:

this matrix directly provides the amounts of basic commodities needed for given amounts of final goods. For example, the bottom left entry of $\backslash mathbf\{AB\}$ is computed as $1\; \backslash cdot\; 1\; +\; 1\; \backslash cdot\; 2\; +\; 2\; \backslash cdot\; 4\; =\; 11$, reflecting that $11$ units of $b\_4$ are needed to produce one unit of $f\_1$. Indeed, one $b\_4$ unit is needed for $m\_1$, one for each of two $m\_2$, and $2$ for each of the four $m\_3$ units that go into the $f\_1$ unit, see picture.

In order to produce e.g. 100 units of the final product $f\_1$, 80 units of $f\_2$, and 60 units of $f\_3$, the necessary amounts of basic goods can be computed as

:$(\backslash mathbf\{AB\})\; \backslash begin\{pmatrix\}\; 100\; \backslash \backslash \; 80\; \backslash \backslash \; 60\; \backslash \backslash \; \backslash end\{pmatrix\}\; =\; \backslash begin\{pmatrix\}\; 1000\; \backslash \backslash \; 1820\; \backslash \backslash \; 1180\; \backslash \backslash \; 2180\; \backslash end\{pmatrix\}\; ,$

that is, $1000$ units of $b\_1$, $1820$ units of $b\_2$, $1180$ units of $b\_3$, $2180$ units of $b\_4$ are needed.

Similarly, the product matrix $\backslash mathbf\{AB\}$ can be used to compute the needed amounts of basic goods for other final-good amount data.{{cite book | isbn=3-446-18668-9 | author=Peter Stingl | title=Mathematik für Fachhochschulen – Technik und Informatik | location=Munich | publisher=Carl Hanser Verlag | edition=5th | year=1996 | language=German }} Here: Exm.5.4.10, p.205-206

The general form of a system of linear equations is

:$\backslash begin\{matrix\}a\_\{11\}x\_1+\backslash cdots\; +\; a\_\{1n\}x\_n=b\_1,$

\\ a_{21}x_1+\cdots + a_{2n}x_n =b_2,

\\ \vdots

\\ a_{m1}x_1+\cdots + a_{mn}x_n =b_m. \end{matrix}

Using same notation as above, such a system is equivalent with the single matrix equation

:$\backslash mathbf\{Ax\}=\backslash mathbf\; b.$

The dot product of two column vectors is the unique entry of the matrix product

:$\backslash mathbf\; x^\backslash mathsf\; T\; \backslash mathbf\; y,$

where $\backslash mathbf\; x^\backslash mathsf\; T$ is the row vector obtained by transposing $\backslash mathbf\; x$. (As usual, a 1×1 matrix is identified with its unique entry.)

More generally, any bilinear form over a vector space of finite dimension may be expressed as a matrix product

:$\backslash mathbf\; x^\backslash mathsf\; T\; \backslash mathbf\; \{Ay\},$

and any sesquilinear form may be expressed as

:$\backslash mathbf\; x^\backslash dagger\; \backslash mathbf\; \{Ay\},$

where $\backslash mathbf\; x^\backslash dagger$ denotes the conjugate transpose of $\backslash mathbf\; x$ (conjugate of the transpose, or equivalently transpose of the conjugate).

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,{{Cite web|last=Weisstein|first=Eric W.|title=Matrix Multiplication|url=https://mathworld.wolfram.com/MatrixMultiplication.html|access-date=2020-09-06|website=mathworld.wolfram.com|language=en}} even when the product remains defined after changing the order of the factors.{{cite book|title=Linear Algebra|edition=4th| first1 = S. | last1 = Lipcshutz | first2 = M. | last2 = Lipson|series=Schaum's Outlines|publisher=McGraw Hill (USA)|chapter=2|date=2009|isbn=978-0-07-154352-1}}{{cite book|title=Matrix Analysis| last = Horn | first = Johnson |edition=2nd |chapter=Chapter 0 |publisher=Cambridge University Press |date=2013|isbn=978-0-521-54823-6}}

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

If {{math|A}} and {{math|B}} are matrices of respective sizes {{tmath|m\times n}} and {{tmath|p\times q}}, then $\backslash mathbf\{A\}\backslash mathbf\{B\}$ is defined if {{tmath|1=n=p}}, and $\backslash mathbf\{B\}\backslash mathbf\{A\}$ is defined if {{tmath|1=m=q}}. Therefore, if one of the products is defined, the other one need not be defined. If {{tmath|1=m=q\neq n=p}}, the two products are defined, but have different sizes; thus they cannot be equal. Only if {{tmath|1=m=q= n=p}}, that is, if {{math|A}} and {{math|B}} are square matrices of the same size, are both products defined and of the same size. Even in this case, one has in general

:$\backslash mathbf\{A\}\backslash mathbf\{B\}\; \backslash neq\; \backslash mathbf\{B\}\backslash mathbf\{A\}.$

For example

:

but

:

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

One special case where commutativity does occur is when {{math|D}} and {{math|E}} are two (square) diagonal matrices (of the same size); then {{math|1=DE = ED}}. Again, if the matrices are over a general ring rather than a field, the corresponding entries in each must also commute with each other for this to hold.

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

:$\backslash mathbf\{A\}(\backslash mathbf\{B\}\; +\; \backslash mathbf\{C\})\; =\; \backslash mathbf\{AB\}\; +\; \backslash mathbf\{AC\},$

and (right distributivity)

:$(\backslash mathbf\{B\}\; +\; \backslash mathbf\{C\}\; )\backslash mathbf\{D\}\; =\; \backslash mathbf\{BD\}\; +\; \backslash mathbf\{CD\}.$

This results from the distributivity for coefficients by

:$\backslash sum\_k\; a\_\{ik\}(b\_\{kj\}\; +\; c\_\{kj\})\; =\; \backslash sum\_k\; a\_\{ik\}b\_\{kj\}\; +\; \backslash sum\_k\; a\_\{ik\}c\_\{kj\}$

:$\backslash sum\_k\; (b\_\{ik\}\; +\; c\_\{ik\})\; d\_\{kj\}\; =\; \backslash sum\_k\; b\_\{ik\}d\_\{kj\}\; +\; \backslash sum\_k\; c\_\{ik\}d\_\{kj\}.$

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

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

:$c(\backslash mathbf\{AB\})\; =\; (c\; \backslash mathbf\{A\})\backslash mathbf\{B\}$ and $(\backslash mathbf\{A\}\; \backslash mathbf\{B\})c=\backslash mathbf\{A\}(\backslash mathbf\{B\}c).$

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

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

:$c\; \backslash left(\backslash sum\_k\; a\_\{ik\}b\_\{kj\}\backslash right)\; =\; \backslash sum\_k\; (c\; a\_\{ik\}\; )\; b\_\{kj\}$

:$\backslash left(\backslash sum\_k\; a\_\{ik\}b\_\{kj\}\backslash right)\; c\; =\; \backslash sum\_k\; a\_\{ik\}\; (\; b\_\{kj\}c).$

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

:$(\backslash mathbf\{AB\})^\backslash mathsf\{T\}\; =\; \backslash mathbf\{B\}^\backslash mathsf\{T\}\backslash mathbf\{A\}^\backslash mathsf\{T\}$

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 {{math|A}} and {{math|B}} is reversed, when one expands the definition of the matrix product.

If {{math|A}} and {{math|B}} have complex entries, then

:$(\backslash mathbf\{AB\})^*\; =\; \backslash mathbf\{A\}^*\backslash mathbf\{B\}^*$

where {{math|^*}} denotes the entry-wise complex conjugate of a matrix.

This results from 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 {{math|A}} and {{math|B}} have complex entries, one has

:$(\backslash mathbf\{AB\})^\backslash dagger\; =\; \backslash mathbf\{B\}^\backslash dagger\backslash mathbf\{A\}^\backslash dagger\; ,$

where {{math|^†}} denotes the conjugate transpose (conjugate of the transpose, or equivalently transpose of the conjugate).

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

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

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

This extends naturally to the product of any number of matrices provided that the dimensions match. That is, if {{math|A₁, A₂, ..., A_n}} are matrices such that the number of columns of {{math|A_i}} equals the number of rows of {{math|A_{i + 1}}} for {{math|1=i = 1, ..., n – 1}}, then the product

:$\backslash prod\_\{i=1\}^n\; \backslash mathbf\{A\}\_i\; =\; \backslash mathbf\{A\}\_1\backslash mathbf\{A\}\_2\backslash cdots\backslash mathbf\{A\}\_n$

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 complicated summation manipulations. This result also follows 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.

Although the result of a sequence of matrix products does not depend 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 {{math|A, B}} and {{math|C}} are matrices of respective sizes {{math|10×30, 30×5, 5×60}}, computing {{math|(AB)C}} needs {{math|1=10×30×5 + 10×5×60 = 4,500}} multiplications, while computing {{math|A(BC)}} needs {{math|1=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 {{mvar|n}} of matrices increases, it has been shown that the choice of the best order has a complexity of $O(n\; \backslash log\; n).${{cite journal

| last1 = Hu

| first1 = T. C. | author1-link = T. C. Hu

| last2 = Shing

| first2 = M.-T.

| title = Computation of Matrix Chain Products, Part I

| journal = SIAM Journal on Computing

| volume = 11

| issue = 2

| pages = 362–373

| year = 1982

| url = http://www.cs.ust.hk/mjg_lib/bibs/DPSu/DPSu.Files/0211028.pdf

| issn = 0097-5397

| doi=10.1137/0211028

| citeseerx = 10.1.1.695.2923

}}

{{cite journal

| last1 = Hu

| first1 = T. C. | author1-link = T. C. Hu

| last2 = Shing

| first2 = M.-T.

| title = Computation of Matrix Chain Products, Part II

| journal = SIAM Journal on Computing

| volume = 13

| issue = 2

| pages = 228–251

| year = 1984

| url = http://www.cs.ust.hk/mjg_lib/bibs/DPSu/DPSu.Files/0213017.pdf

| issn = 0097-5397

| doi=10.1137/0213017

| citeseerx = 10.1.1.695.4875

}}

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

:$S\_\backslash mathbf\{P\}(\backslash mathbf\{A\})\; =\; \backslash mathbf\{P\}^\{-1\}\; \backslash mathbf\{A\}\; \backslash mathbf\{P\}.$

Similarity transformations map product to products, that is

:$S\_\backslash mathbf\{P\}(\backslash mathbf\{AB\})\; =\; S\_\backslash mathbf\{P\}(\backslash mathbf\{A\})S\_\backslash mathbf\{P\}(\backslash mathbf\{B\}).$

In fact, one has

:$\backslash mathbf\{P\}^\{-1\}\; (\backslash mathbf\{AB\})\; \backslash mathbf\{P\}$

= \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}).

Let us denote $\backslash mathcal\; M\_n(R)$ the set of {{math|n×n}} square matrices with entries in a ring {{mvar|R}}, which, in practice, is often a field.

In $\backslash mathcal\; M\_n(R)$, the product is defined for every pair of matrices. This makes $\backslash mathcal\; M\_n(R)$ a ring, which has the identity matrix {{math|I}} as identity element (the matrix whose diagonal entries are equal to 1 and all other entries are 0). This ring is also an associative algebra.

If {{math|n > 1}}, many matrices do not have a multiplicative inverse. For example, a matrix such that all entries of a row (or a column) are 0 does not have an inverse. If it exists, the inverse of a matrix {{math|A}} is denoted {{math|A⁻¹}}, and, thus verifies

:$\backslash mathbf\{A\}\backslash mathbf\{A\}^\{-1\}\; =\; \backslash mathbf\{A\}^\{-1\}\backslash mathbf\{A\}\; =\; \backslash mathbf\{I\}.$

A matrix that has an inverse is an invertible matrix. Otherwise, it is a singular matrix.

A product of matrices is invertible if and only if each factor is invertible. In this case, one has

:$(\backslash mathbf\{A\}\backslash mathbf\{B\})^\{-1\}\; =\; \backslash mathbf\{B\}^\{-1\}\backslash mathbf\{A\}^\{-1\}.$

When {{mvar|R}} is commutative, and, in particular, when it is a field, the determinant of a product is the product of the determinants. As determinants are scalars, and scalars commute, one has thus

:$\backslash det(\backslash mathbf\{AB\})\; =\; \backslash det(\backslash mathbf\{BA\})\; =\backslash det(\backslash mathbf\{A\})\backslash det(\backslash mathbf\{B\}).$

The other matrix invariants do not behave as well with products. Nevertheless, if {{mvar|R}} is commutative, {{math|AB}} and {{math|BA}} have the same trace, the same characteristic polynomial, and the same eigenvalues with the same multiplicities. However, the eigenvectors are generally different if {{math|AB ≠ BA}}.

One may raise a square matrix to any nonnegative integer power multiplying it by itself repeatedly in the same way as for ordinary numbers. That is,

:$\backslash mathbf\{A\}^0\; =\; \backslash mathbf\{I\},$

:$\backslash mathbf\{A\}^1\; =\; \backslash mathbf\{A\},$

:$\backslash mathbf\{A\}^k\; =\; \backslash underbrace\{\backslash mathbf\{A\}\backslash mathbf\{A\}\backslash cdots\backslash mathbf\{A\}\}\_\{k\backslash text\{\; times\}\}.$

Computing the {{mvar|k}}th power of a matrix needs {{math|k – 1}} times the time of a single matrix multiplication, if it is done with the trivial algorithm (repeated multiplication). As this may be very time consuming, one generally prefers using exponentiation by squaring, which requires less than {{math|2 log₂ k}} matrix multiplications, and is therefore much more efficient.

An easy case for exponentiation is that of a diagonal matrix. Since the product of diagonal matrices amounts to simply multiplying corresponding diagonal elements together, the {{mvar|k}}th power of a diagonal matrix is obtained by raising the entries to the power {{mvar|k}}:

:$$

\begin{bmatrix}

a_{11} & 0 & \cdots & 0 \\

0 & a_{22} & \cdots & 0 \\

\vdots & \vdots & \ddots & \vdots \\

0 & 0 & \cdots & a_{nn}

\end{bmatrix}^k =

\begin{bmatrix}

a_{11}^k & 0 & \cdots & 0 \\

0 & a_{22}^k & \cdots & 0 \\

\vdots & \vdots & \ddots & \vdots \\

0 & 0 & \cdots & a_{nn}^k

\end{bmatrix}.

The definition of matrix product requires that the entries belong to a semiring, and does not require multiplication of elements of the semiring to be commutative. In many applications, the matrix elements belong to a field, although the tropical semiring is also a common choice for graph shortest path problems.{{cite book|title=Randomized Algorithms|first1=Rajeev|last1=Motwani|author1-link=Rajeev Motwani|first2=Prabhakar|last2=Raghavan|author2-link=Prabhakar Raghavan|publisher=Cambridge University Press|year=1995|isbn=9780521474658|page=280|url=https://books.google.com/books?id=QKVY4mDivBEC&pg=PA280}} Even in the case of matrices over fields, the 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 entries are zero outside of the main diagonal and 1 on the main diagonal) are identity elements of the matrix product. It follows that the {{math|n × n}} matrices over a ring form a ring, which is noncommutative except if {{math|1=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 {{mvar|R}}, a matrix has an inverse if and only if its determinant has a multiplicative inverse in {{mvar|R}}. The determinant of a product of square matrices is the product of the determinants of the factors. The {{math|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.

Matrices are the morphisms of a category, the category of matrices. The objects are the natural numbers that measure the size of matrices, and the composition of morphisms is matrix multiplication. The source of a morphism is the number of columns of the corresponding matrix, and the target is the number of rows.

{{Main|Computational complexity of matrix multiplication}}

{{For|implementation techniques (in particular parallel and distributed algorithms)|Matrix multiplication algorithm}}

File:MatrixMultComplexity svg.svg

The matrix multiplication algorithm that results from the definition requires, in the worst case, {{tmath|n^3}} multiplications and {{tmath|(n-1)n^2}} additions of scalars to compute the product of two square {{math|n×n}} matrices. Its computational complexity is therefore {{tmath|O(n^3)}}, in a model of computation for which the scalar operations take constant time.

Rather surprisingly, this complexity is not optimal, as shown in 1969 by Volker Strassen, who provided an algorithm, now called Strassen's algorithm, with a complexity of $O(\; n^\{\backslash log\_\{2\}7\})\; \backslash approx\; O(n^\{2.8074\}).$

{{cite journal

| doi=10.1007/BF02165411

| url=http://www.digizeitschriften.de/dms/img/?PID=GDZPPN001168215

| author=Volker Strassen

| title=Gaussian elimination is not optimal

| journal=Numerische Mathematik

| volume=13

| pages=354–356

| date=Aug 1969

| issue = 4

| s2cid = 121656251

}}

Strassen's algorithm can be parallelized to further improve the performance.{{cite journal | url=https://core.ac.uk/download/pdf/82778592.pdf | author=C.-C. Chou and Y.-F. Deng and G. Li and Y. Wang | title=Parallelizing Strassen's Method for Matrix Multiplication on Distributed-Memory MIMD Architectures | journal=Computers Math. Applic. | volume=30 | number=2 | pages=49–69 | year=1995 | doi=10.1016/0898-1221(95)00077-C }}

{{As of|2024|01}}, the best peer-reviewed matrix multiplication algorithm is by Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, and Renfei Zhou and has complexity {{math|O(n^2.371552)}}.{{cite conference |last1=Vassilevska Williams |first1=Virginia |last2=Xu |first2=Yinzhan |last3=Xu |first3=Zixuan |last4=Zhou |first4=Renfei |title=New Bounds for Matrix Multiplication: from Alpha to Omega |conference=Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) |pages=3792–3835 |arxiv=2307.07970 |doi=10.1137/1.9781611977912.134}}{{cite web |last=Nadis |first=Steve |date=March 7, 2024 |title=New Breakthrough Brings Matrix Multiplication Closer to Ideal |url=https://www.quantamagazine.org/new-breakthrough-brings-matrix-multiplication-closer-to-ideal-20240307 |access-date=2024-03-09}}

It is not known whether matrix multiplication can be performed in {{math|n^{2 + o(1)}}} time.that is, in time {{math|n^2+f(n)}}, for some function {{mvar|f}} with {{math|f(n)→0}} as {{math|n→∞}} This would be optimal, since one must read the {{tmath|n^2}} elements of a matrix in order to multiply it with another matrix.

Since matrix multiplication forms the basis for many algorithms, and many operations on matrices even have the same complexity as matrix multiplication (up to a multiplicative constant), the computational complexity of matrix multiplication appears throughout numerical linear algebra and theoretical computer science.

Other types of products of matrices include:

• Block matrix operations
• Cracovian product, defined as {{math|1=A ∧ B = B^TA}}
• Frobenius inner product, the dot product of matrices considered as vectors, or, equivalently the sum of the entries of the Hadamard product
• Hadamard product of two matrices of the same size, resulting in a matrix of the same size, which is the product entry-by-entry
• Kronecker product or tensor product, the generalization to any size of the preceding
• Khatri–Rao product and face-splitting product
• Outer product, also called dyadic product or tensor product of two column matrices, which is $\backslash mathbf\{a\}\backslash mathbf\{b\}^\backslash mathsf\{T\}$
• Scalar multiplication

• Matrix calculus, for the interaction of matrix multiplication with operations from calculus

{{Reflist|30em}}

{{Commons category|Matrix multiplication|matrix multiplication}}

{{wikibooks

|1= Linear Algebra

|2= Matrix Multiplication

|3= Matrix multiplication

}}

{{wikibooks|Applicable Mathematics|Matrices#Multiplying Matrices|Multiplying Matrices}}

{{refbegin}}

• 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.
• {{cite journal | last1 = Coppersmith | first1 = D. | last2 = Winograd | first2 = S. | year = 1990 | title = Matrix multiplication via arithmetic progressions | journal = J. Symbolic Comput. | volume = 9 | issue = 3| pages = 251–280 | doi=10.1016/s0747-7171(08)80013-2| doi-access = free }}
• {{Citation | last1=Horn | first1=Roger A. | last2=Johnson | first2=Charles R. | title=Topics in Matrix Analysis | publisher=Cambridge University Press | isbn=978-0-521-46713-1 | year=1991}}
• 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.
• {{Citation | last1=Press | first1=William H. | last2=Flannery | first2=Brian P. | last3=Teukolsky | first3=Saul A. | author3-link=Saul Teukolsky | last4=Vetterling | first4=William T. | title=Numerical Recipes: The Art of Scientific Computing | publisher=Cambridge University Press | edition=3rd | isbn=978-0-521-88068-8 | year=2007| title-link=Numerical Recipes }}.
• 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. [https://web.archive.org/web/20100331095603/http://www.siam.org/pdf/news/174.pdf PDF]
• Strassen, Volker, Gaussian Elimination is not Optimal, Numer. Math. 13, p. 354–356, 1969.
• {{Citation | doi=10.1016/0024-3795(73)90023-2 | last=Styan | first=George P. H. | title=Hadamard Products and Multivariate Statistical Analysis | journal=Linear Algebra and Its Applications | year=1973 | volume=6 | pages=217–240| url=http://dml.cz/bitstream/handle/10338.dmlcz/102190/CzechMathJ_37-1987-4_14.pdf | doi-access=free }}
• {{Cite book|last=Williams|first=Virginia Vassilevska|date=2012-05-19|chapter=Multiplying matrices faster than coppersmith-winograd|chapter-url=http://dl.acm.org/citation.cfm?id=2213977.2214056|publisher=ACM|pages=887–898|doi=10.1145/2213977.2214056|isbn=9781450312455|title=Proceedings of the 44th symposium on Theory of Computing - STOC '12|citeseerx=10.1.1.297.2680|s2cid=14350287}}

{{refend}}

{{Linear algebra}}

Category:Matrix theory

Category:Bilinear maps

Category:Multiplication

Category:Numerical linear algebra