Matrix multiplication

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.

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

The values at the intersections marked with circles are:

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

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 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 A from a vector space of dimension n into a vector space of dimension m maps a column vector

${\displaystyle \mathbf {x} ={\begin{pmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{pmatrix}}}$

onto the column vector

${\displaystyle \mathbf {y} =A(\mathbf {x} )={\begin{pmatrix}a_{11}x_{1}+\cdots +a_{1n}x_{n}\\a_{21}x_{1}+\cdots +a_{2n}x_{n}\\\vdots \\a_{m1}x_{1}+\cdots +a_{mn}x_{n}\end{pmatrix}}.}$

The linear map A is thus defined by the matrix

${\displaystyle \mathbf {A} ={\begin{pmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\\\end{pmatrix}},}$

and maps the column vector ${\displaystyle \mathbf {x} }$ to the matrix product

${\displaystyle \mathbf {y} =\mathbf {Ax} .}$

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

${\displaystyle (\mathbf {BA} )\mathbf {x} =\mathbf {B} (\mathbf {Ax} )=\mathbf {BAx} .}$

Using a Cartesian coordinate system in a Euclidean plane, the rotation by an angle ${\displaystyle \alpha }$ around the origin is a linear map. More precisely,

${\displaystyle {\begin{bmatrix}x'\\y'\end{bmatrix}}={\begin{bmatrix}\cos \alpha &-\sin \alpha \\\sin \alpha &\cos \alpha \end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}}$,

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

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

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

where appropriate trigonometric identities are employed for the second equality. That is, the composition corresponds to the rotation by angle ${\displaystyle \alpha +\beta }$, as expected.

The computation of the bottom left entry of ${\displaystyle \mathbf {AB} }$ corresponds to the consideration of all paths (highlighted) from basic commodity ${\displaystyle b_{4}}$ to final product ${\displaystyle f_{1}}$ in the production flow graph.

As an example, a fictitious factory uses 4 kinds of basic commodities, ${\displaystyle b_{1},b_{2},b_{3},b_{4}}$ to produce 3 kinds of intermediate goods, ${\displaystyle m_{1},m_{2},m_{3}}$, which in turn are used to produce 3 kinds of final products, ${\displaystyle f_{1},f_{2},f_{3}}$. The matrices

${\displaystyle \mathbf {A} =\left({\begin{array}{ccc}1&0&1\\2&1&1\\0&1&1\\1&1&2\\\end{array}}\right)}$   and   ${\displaystyle \mathbf {B} =\left({\begin{array}{ccc}1&2&1\\2&3&1\\4&2&2\\\end{array}}\right)}$

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 ${\displaystyle m_{1}}$, one unit of basic commodity ${\displaystyle b_{1}}$, two units of ${\displaystyle b_{2}}$, no units of ${\displaystyle b_{3}}$, and one unit of ${\displaystyle b_{4}}$ are needed, corresponding to the first column of ${\displaystyle \mathbf {A} }$.

Using matrix multiplication, compute

${\displaystyle \mathbf {AB} =\left({\begin{array}{ccc}5&4&3\\8&9&5\\\ 6&5&3\\11&9&6\\\end{array}}\right)}$;

this matrix directly provides the amounts of basic commodities needed for given amounts of final goods. For example, the bottom left entry of ${\displaystyle \mathbf {AB} }$ is computed as ${\displaystyle 1\cdot 1+1\cdot 2+2\cdot 4=11}$, reflecting that ${\displaystyle 11}$ units of ${\displaystyle b_{4}}$ are needed to produce one unit of ${\displaystyle f_{1}}$. Indeed, one ${\displaystyle b_{4}}$ unit is needed for ${\displaystyle m_{1}}$, 2 for ${\displaystyle m_{2}}$, and ${\displaystyle 4}$ for each of the two ${\displaystyle m_{3}}$ units that go into the ${\displaystyle f_{1}}$ unit, see picture.

In order to produce e.g. 100 units of the final product ${\displaystyle f_{1}}$, 80 units of ${\displaystyle f_{2}}$, and 60 units of ${\displaystyle f_{3}}$, the necessary amounts of basic goods can be computed as

${\displaystyle (\mathbf {AB} )\left({\begin{array}{c}100\\80\\60\\\end{array}}\right)=\left({\begin{array}{c}1000\\1820\\1180\\2180\\\end{array}}\right)}$,

that is, ${\displaystyle 1000}$ units of ${\displaystyle b_{1}}$, ${\displaystyle 1820}$ units of ${\displaystyle b_{2}}$, ${\displaystyle 1180}$ units of ${\displaystyle b_{3}}$, ${\displaystyle 2180}$ units of ${\displaystyle b_{4}}$ are needed. Similarly, the product matrix ${\displaystyle \mathbf {AB} }$ can be used to compute the needed amounts of basic goods for other final-good amount data.[9]

The general form of a system of linear equations is

${\displaystyle {\begin{matrix}a_{11}x_{1}+\cdots +a_{1n}x_{n}=b_{1}\\a_{21}x_{1}+\cdots +a_{2n}x_{n}=b_{2}\\\vdots \\a_{m1}x_{1}+\cdots +a_{mn}x_{n}=b_{m}\end{matrix}}.}$

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

${\displaystyle \mathbf {Ax} =\mathbf {b} .}$

The dot product of two column vectors is the matrix product

${\displaystyle \mathbf {x} ^{\mathsf {T}}\mathbf {y} ,}$

where ${\displaystyle \mathbf {x} ^{\mathsf {T}}}$ is the row vector obtained by transposing ${\displaystyle \mathbf {x} }$ and the resulting 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

${\displaystyle \mathbf {x} ^{\mathsf {T}}\mathbf {Ay} ,}$

and any sesquilinear form may be expressed as

${\displaystyle \mathbf {x} ^{\dagger }\mathbf {Ay} ,}$

where ${\displaystyle \mathbf {x} ^{\dagger }}$ denotes the conjugate transpose of ${\displaystyle \mathbf {x} }$ (conjugate of the transpose, or equivalently transpose of the conjugate).

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 one need not be defined. If ${\displaystyle m=q\neq n=p}$, the two products are defined, but have different sizes; thus they cannot be equal. Only if ${\displaystyle m=q=n=p}$, that is, if A and 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

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

For example

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

but

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

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 a ring, then one must add the condition that c belongs to the center of the ring.

One special case where commutativity does occur is when D and E are two (square) diagonal matrices (of the same size); then DE = ED.[10] 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 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)

${\displaystyle (\mathbf {B} +\mathbf {C} )\mathbf {D} =\mathbf {BD} +\mathbf {CD} .}$[10]

This results from the distributivity for coefficients by

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

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

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 or 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:

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

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

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.

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 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 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).

Given three matrices A, B and C, the products (AB)C and A(BC) are defined if and only if 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 products is defined, then 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 matrices provided that the dimensions match. That is, if A₁, A₂, ..., A_n are matrices such that the number of columns of A_i equals the number of rows of A_{i + 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 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 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 ${\displaystyle O(n\log n).}$

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

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

In fact, one has

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

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,

${\displaystyle \mathbf {A} ^{0}=\mathbf {I} ,}$

${\displaystyle \mathbf {A} ^{1}=\mathbf {A} ,}$

${\displaystyle \mathbf {A} ^{k}=\underbrace {\mathbf {A} \mathbf {A} \cdots \mathbf {A} } _{k{\text{ times}}}.}$

Computing the kth power of a matrix needs 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 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 kth power of a diagonal matrix is obtained by raising the entries to the power k:

${\displaystyle {\begin{bmatrix}a_{11}&0&\cdots &0\\0&a_{22}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &a_{nn}\end{bmatrix}}^{k}={\begin{bmatrix}a_{11}^{k}&0&\cdots &0\\0&a_{22}^{k}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &a_{nn}^{k}\end{bmatrix}}.}$