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.
Linear maps 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 from a vector space of dimension into a vector space of dimension maps a column vector :\mathbf x=\begin{pmatrix}x_1 \\ x_2 \\ \vdots \\ x_n\end{pmatrix} onto the column vector :\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 is thus defined by the matrix :\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 \mathbf x to the matrix product :\mathbf y = \mathbf {Ax}. If is another linear map from the preceding vector space of dimension , into a vector space of dimension , it is represented by a matrix \mathbf B. A straightforward computation shows that the matrix of the
composite map is the matrix product \mathbf {BA}. The general formula ) that defines the function composition is instanced here as a specific case of associativity of matrix product (see below): :(\mathbf{BA})\mathbf x = \mathbf{B}(\mathbf {Ax}) = \mathbf{BAx}.
Geometric rotations Using a
Cartesian coordinate system in a Euclidean plane, the
rotation by an angle \alpha around the
origin is a linear map. More precisely, \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 (x,y) and its image (x',y') are written as column vectors. The composition of the rotation by \alpha and that by \beta then corresponds to the matrix product \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 \alpha+\beta, as expected.
Resource allocation in economics 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 :\mathbf{A} = \begin{pmatrix} 1 & 0 & 1 \\ 2 & 1 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 2 \\ \end{pmatrix} and \mathbf{B} = \begin{pmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 4 & 2 & 2 \\ \end{pmatrix} 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 \mathbf{A}. Using matrix multiplication, compute :\mathbf{AB} = \begin{pmatrix} 5 & 4 & 3 \\ 8 & 9 & 5 \\\ 6 & 5 & 3 \\ 11 & 9 & 6 \\ \end{pmatrix} ; this matrix directly provides the amounts of basic commodities needed for given amounts of final goods. For example, the bottom left entry of \mathbf{AB} is computed as 1 \cdot 1 + 1 \cdot 2 + 2 \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 :(\mathbf{AB}) \begin{pmatrix} 100 \\ 80 \\ 60 \\ \end{pmatrix} = \begin{pmatrix} 1000 \\ 1820 \\ 1180 \\ 2180 \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 \mathbf{AB} can be used to compute the needed amounts of basic goods for other final-good amount data.
System of linear equations The general form of a
system of linear equations is :\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 :\mathbf{Ax}=\mathbf b.
Dot product, bilinear form and sesquilinear form The
dot product of two column vectors is the unique entry of the matrix product :\mathbf x^\mathsf T \mathbf y, where \mathbf x^\mathsf T is the
row vector obtained by
transposing \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 :\mathbf x^\mathsf T \mathbf {Ay}, and any
sesquilinear form may be expressed as :\mathbf x^\dagger \mathbf {Ay}, where \mathbf x^\dagger denotes the
conjugate transpose of \mathbf x (conjugate of the transpose, or equivalently transpose of the conjugate). ==General properties==