Editing Matrix multiplication (section)

==Definitions==
===Matrix times matrix===
If {{math|'''A'''}} is an {{math|''m'' × ''n''}} matrix and {{math|'''B'''}} is an {{math|''n'' × ''p''}} matrix,
<math display="block">\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},\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}</math>
the ''matrix product'' {{math|1='''C''' = '''AB'''}} (denoted without multiplication signs or dots) is defined to be the {{math|''m'' × ''p''}} matrix<ref>{{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}}</ref><ref>{{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}}</ref><ref>{{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}}</ref><ref>{{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}}</ref>
<math display="block">\mathbf{C} = \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}</math>
such that 
<math display="block"> c_{ij} = a_{i1} b_{1j} + a_{i2} b_{2j} + \cdots + a_{in} b_{nj} = \sum_{k=1}^n a_{ik} b_{kj}, </math>
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
<math display="block">\mathbf{C} = \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} 
</math>

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'''}},<ref name=":1" /> in this case {{math|''n''}}.

In most scenarios, the entries are numbers, but they may be any kind of [[mathematical object]]s for which an addition and a multiplication are defined, that are [[associative property|associative]], and such that the addition is [[commutative property|commutative]], and the multiplication is [[distributive property|distributive]] with respect to the addition. In particular, the entries may be matrices themselves (see [[block matrix]]).

===Matrix times vector===
A vector <math>\mathbf x</math> of length <math>n</math> can be viewed as a [[column vector]], corresponding to an <math>n\times1</math> matrix <math>\mathbf X</math> whose entries are given by <math>\mathbf X_{i1}=\mathbf x_i.</math> If <math>\mathbf A</math> is an <math>m\times n</math> matrix, the matrix-times-vector product denoted by <math>\mathbf {Ax}</math> is then the vector <math>\mathbf y</math> that, viewed as a column vector, is equal to the <math>m\times1</math> matrix <math>\mathbf{AX}.</math> In index notation, this amounts to:
:<math>y_i=\sum_{j=1}^n a_{ij}x_j.</math>
One way of looking at this is that the changes from "plain" vector to column vector and back are assumed and left implicit.

===Vector times matrix===
Similarly, a vector <math>\mathbf x</math> of length <math>n</math> can be viewed as a [[row vector]], corresponding to a <math>1\times n</math> 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 <math>\mathbf{x}^\mathrm{T}\mathbf{A}.</math> The identity <math>\mathbf{x}^\mathrm{T}\mathbf{A}=(\mathbf{A}^\mathrm{T}\mathbf{x})^\mathrm{T}</math> holds. In index notation, if <math>\mathbf A</math> is an <math>n\times p</math> matrix, <math>\mathbf{x}^\mathrm{T}\mathbf{A}=\mathbf{y}^\mathrm{T}</math> amounts to:
<math>y_k=\sum_{j=1}^n x_j a_{jk}.</math>

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

===Illustration===
[[File:Matrix multiplication diagram 2.svg|right|thumb]]

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'''}}.
<math display="block">
\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}}
</math>

The values at the intersections, marked with circles in figure to the right, are:
<math display="block">\begin{align}
c_{12} & = a_{11} b_{12} + a_{12} b_{22} \\
c_{33} & = a_{31} b_{13} + a_{32} b_{23} .
\end{align}</math>