Editing Matrix multiplication (section)

==Fundamental applications==
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 (linear algebra)|basis]], its vectors are each uniquely represented by a finite [[sequence (mathematics)|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 [[isomorphism|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
:<math>\mathbf x=\begin{pmatrix}x_1 \\ x_2 \\ \vdots \\ x_n\end{pmatrix}</math>
onto the column vector
:<math>\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}.</math>
The linear map {{mvar|A}} is thus defined by the matrix 
:<math>\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}, </math>
and maps the column vector <math>\mathbf x</math> to the matrix product 
:<math>\mathbf y = \mathbf {Ax}.</math>

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 <math>\mathbf B.</math> A straightforward computation shows that the matrix of the [[function composition|composite map]] {{tmath|B\circ A}} is the matrix product <math>\mathbf {BA}.</math> 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):
:<math>(\mathbf{BA})\mathbf x = \mathbf{B}(\mathbf {Ax}) = \mathbf{BAx}.</math>

====Geometric rotations====
{{See also|Rotation matrix}}
Using a [[Cartesian coordinate]] system in a Euclidean plane, the [[rotation (mathematics)|rotation]] by an angle <math>\alpha</math> around the [[origin (mathematics)|origin]] is a linear map.
More precisely, 
<math display="block"> \begin{bmatrix} x' \\ y' \end{bmatrix} =
 \begin{bmatrix} \cos \alpha & - \sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix},</math>
where the source point <math>(x,y)</math> and its image <math>(x',y')</math> are written as column vectors.

The composition of the rotation by <math>\alpha</math> and that by <math>\beta</math> then corresponds to the matrix product
<math display="block">\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},</math>
where appropriate [[List of trigonometric identities#Angle sum and difference identities|trigonometric identities]] are employed for the second equality.
That is, the composition corresponds to the rotation by angle <math>\alpha+\beta</math>, as expected.

====Resource allocation in economics====
[[File:Mmult factory svg.svg|thumb|400px|The computation of the bottom left entry of <math>\mathbf{AB}</math> corresponds to the consideration of all paths (highlighted) from basic commodity <math>b_4</math> to final product <math>f_1</math> in the production flow graph.]]
As an example, a fictitious factory uses 4 kinds of [[primary commodity|basic commodities]], <math>b_1, b_2, b_3, b_4</math> to produce 3 kinds of [[intermediate good]]s, <math>m_1, m_2, m_3</math>, which in turn are used to produce 3 kinds of [[final product]]s, <math>f_1, f_2, f_3</math>. The matrices 
:<math>\mathbf{A} = \begin{pmatrix} 1 & 0 & 1 \\   2 & 1 & 1 \\   0 & 1 & 1 \\   1 & 1 & 2 \\   \end{pmatrix} </math> &nbsp; and &nbsp; <math>\mathbf{B} = \begin{pmatrix} 1 & 2 & 1 \\   2 & 3 & 1 \\   4 & 2 & 2 \\   \end{pmatrix} </math>
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 <math>m_1</math>, one unit of basic commodity <math>b_1</math>, two units of <math>b_2</math>, no units of <math>b_3</math>, and one unit of <math>b_4</math> are needed, corresponding to the first column of <math>\mathbf{A}</math>.

Using matrix multiplication, compute 
:<math>\mathbf{AB} = \begin{pmatrix} 5 & 4 & 3 \\   8 & 9 & 5 \\\   6 & 5 & 3 \\   11 & 9 & 6 \\   \end{pmatrix} ;</math>
this matrix directly provides the amounts of basic commodities needed for given amounts of final goods. For example, the bottom left entry of <math>\mathbf{AB}</math> is computed as <math>1 \cdot 1 + 1 \cdot 2 + 2 \cdot 4 = 11</math>, reflecting that <math>11</math> units of <math>b_4</math> are needed to produce one unit of <math>f_1</math>. Indeed, one <math>b_4</math> unit is needed for <math>m_1</math>, one for each of two <math>m_2</math>, and <math>2</math> for each of the four <math>m_3</math> units that go into the <math>f_1</math> unit, see picture.

In order to produce e.g. 100 units of the final product <math>f_1</math>, 80 units of <math>f_2</math>, and 60 units of <math>f_3</math>, the necessary amounts of basic goods can be computed as
:<math>(\mathbf{AB}) \begin{pmatrix} 100 \\ 80 \\ 60 \\ \end{pmatrix} = \begin{pmatrix} 1000 \\ 1820 \\ 1180 \\ 2180 \end{pmatrix} ,</math>
that is, <math>1000</math> units of <math>b_1</math>, <math>1820</math> units of <math>b_2</math>, <math>1180</math> units of <math>b_3</math>, <math>2180</math> units of <math>b_4</math> are needed.
Similarly, the product matrix <math>\mathbf{AB}</math> can be used to compute the needed amounts of basic goods for other final-good amount data.<ref>{{cite book | isbn=3-446-18668-9 | author=Peter Stingl | title=Mathematik für Fachhochschulen &ndash; Technik und Informatik | location=[[Munich]] | publisher=[[Carl Hanser Verlag]] | edition=5th | year=1996 | language=German }} Here: Exm.5.4.10, p.205-206</ref>

===System of linear equations===
The general form of a [[system of linear equations]] is
:<math>\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}</math>

Using same notation as above, such a system is equivalent with the single matrix [[equation]]
:<math>\mathbf{Ax}=\mathbf b.</math>

===Dot product, bilinear form and sesquilinear form===
The [[dot product]] of two column vectors is the unique entry of the matrix product 
:<math>\mathbf x^\mathsf T \mathbf y,</math>
where <math>\mathbf x^\mathsf T</math> is the [[row vector]] obtained by [[transpose|transposing]] <math>\mathbf x</math>.  (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
:<math>\mathbf x^\mathsf T \mathbf {Ay},</math>
and any [[sesquilinear form]] may be expressed as 
:<math>\mathbf x^\dagger \mathbf {Ay},</math>
where <math>\mathbf x^\dagger</math> denotes the [[conjugate transpose]] of <math>\mathbf x</math> (conjugate of the transpose, or equivalently transpose of the conjugate).