Difference between revisions of "Matrix"
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A '''matrix''' (pl.: "matrices," [[Latin]] origin) is a complex ordering, in deliberate fashion, of [[numeral|numerals]]. In [[mathematics]], a "matrix" is a regular grid of numbers, which may be manipulated and solved through intermediate-level [[algebra]]. Matrix algebra is usually taught in [[sophomore]] [[high school]] level mathematics. | A '''matrix''' (pl.: "matrices," [[Latin]] origin) is a complex ordering, in deliberate fashion, of [[numeral|numerals]]. In [[mathematics]], a "matrix" is a regular grid of numbers, which may be manipulated and solved through intermediate-level [[algebra]]. Matrix algebra is usually taught in [[sophomore]] [[high school]] level mathematics. | ||
| + | |||
| + | More formally, a matrix is an example of a rank-2 [[tensor]]. | ||
Alternately, a '''matrix''' may also be a complex ordering of a group of equivalent objects, especially where the order is imposed to gain incidental benefit from the [[synergy]] of the networked objects. | Alternately, a '''matrix''' may also be a complex ordering of a group of equivalent objects, especially where the order is imposed to gain incidental benefit from the [[synergy]] of the networked objects. | ||
==Mathematics== | ==Mathematics== | ||
| − | In mathematics, matrices can be manipulated in a variety of ways, including [[addition]] and multiplication. For example, to add | + | In mathematics, matrices can be manipulated in a variety of ways, including [[addition]] and multiplication. |
| + | ===Addition of matrices=== | ||
| + | For example, to add two matrices, one would add their respective elements, thus: | ||
<math>\begin{bmatrix} | <math>\begin{bmatrix} | ||
| Line 27: | Line 31: | ||
0 & 6 & v | 0 & 6 & v | ||
\end{bmatrix} </math> | \end{bmatrix} </math> | ||
| + | |||
| + | ===Multiplication of matrices=== | ||
| + | To multiply two matrices, one uses the rule "go along the rows and down the columns". This is best illustrated by a specific example: a matrix times a vector: | ||
| + | |||
| + | <math>\begin{bmatrix} | ||
| + | a & b \\ | ||
| + | c & d | ||
| + | \end{bmatrix} \begin{bmatrix} | ||
| + | e \\ f | ||
| + | \end{bmatrix}=\begin{bmatrix} | ||
| + | ae+bf \\ | ||
| + | ce+df \end{bmatrix} | ||
| + | </math> | ||
| + | |||
| + | It is important to note that matrix multiplication is not commutative: in general, <math>AB \neq BA</math> for two matrices <math>A</math> and <math>B</math>. This has important consequences in [[quantum mechanics]]. | ||
| + | |||
| + | To see why matrix multiplication works the way it does, we will use [[suffix notation]]. Consider first forming the product of two matrices, <math>AB</math>, which is itself a matrix. Then form the product <math>ABx</math>. Matrix multiplication is associative, so we can consider this as either <math>(AB)x</math> or <math>A(Bx)</math>. In suffix notation, | ||
| + | |||
| + | <math> | ||
| + | \sum_j(AB)_{ij}x_j = \sum_k A_{ik}(Bx)_k=\sum_{j,k}A_{ik}B_{kj}x_j | ||
| + | </math> | ||
| + | The vector <math>x</math> is arbitrary, so we can therefore deduce the rule for finding the product of two matrices: | ||
| + | |||
| + | <math> | ||
| + | (AB)_{ij}=\sum_k A_{ik}B_{kj} | ||
| + | </math> | ||
[[Category:Mathematics]] | [[Category:Mathematics]] | ||
[[Category:Computers]] | [[Category:Computers]] | ||
Revision as of 21:09, May 11, 2008
A matrix (pl.: "matrices," Latin origin) is a complex ordering, in deliberate fashion, of numerals. In mathematics, a "matrix" is a regular grid of numbers, which may be manipulated and solved through intermediate-level algebra. Matrix algebra is usually taught in sophomore high school level mathematics.
More formally, a matrix is an example of a rank-2 tensor.
Alternately, a matrix may also be a complex ordering of a group of equivalent objects, especially where the order is imposed to gain incidental benefit from the synergy of the networked objects.
Mathematics
In mathematics, matrices can be manipulated in a variety of ways, including addition and multiplication.
Addition of matrices
For example, to add two matrices, one would add their respective elements, thus:
would equal
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} {x+0} & {y+3} & {z+1} \\ {1+4} & {3+3} & {5+(x+2)} \\ {0+0} & {2+4} & {0+v} \end{bmatrix} = \begin{bmatrix} x & {y+3} & {z+1} \\ 5 & 6 & {x+7} \\ 0 & 6 & v \end{bmatrix} }
Multiplication of matrices
To multiply two matrices, one uses the rule "go along the rows and down the columns". This is best illustrated by a specific example: a matrix times a vector:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} e \\ f \end{bmatrix}=\begin{bmatrix} ae+bf \\ ce+df \end{bmatrix} }
It is important to note that matrix multiplication is not commutative: in general, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AB \neq BA} for two matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} . This has important consequences in quantum mechanics.
To see why matrix multiplication works the way it does, we will use suffix notation. Consider first forming the product of two matrices, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AB} , which is itself a matrix. Then form the product Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ABx} . Matrix multiplication is associative, so we can consider this as either Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (AB)x} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(Bx)} . In suffix notation,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_j(AB)_{ij}x_j = \sum_k A_{ik}(Bx)_k=\sum_{j,k}A_{ik}B_{kj}x_j } The vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is arbitrary, so we can therefore deduce the rule for finding the product of two matrices:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (AB)_{ij}=\sum_k A_{ik}B_{kj} }