Difference between revisions of "Matrix"

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''For the 1999 film, see [[The Matrix]].''
 
''For the 1999 film, see [[The 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.
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A '''matrix''' (pl.: "matrices," [[Latin]] origin) is a complex ordering, in deliberate fashion, of [[numeral]]s.  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]].
 
More formally, a matrix is an example of a rank-2 [[tensor]].
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===Addition of matrices===
 
===Addition of matrices===
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For example, to add two matrices, one would add their respective elements, thus:
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To add two matrices, one would add their respective elements.  For example:
  
 
<math>\begin{bmatrix}
 
<math>\begin{bmatrix}
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===Multiplication of matrices===
 
===Multiplication of matrices===
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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:
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To multiply two matrices, use the rule for finding the product of two matrices:
  
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<math>\begin{bmatrix}
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<math>
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a & b \\
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(AB)_{ij}=\sum_k A_{ik}B_{kj}
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c & d
 
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\end{bmatrix} \begin{bmatrix}
 
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e \\ f
 
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\end{bmatrix}=\begin{bmatrix}
 
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ae+bf \\
 
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ce+df \end{bmatrix}
 
 
</math>
 
</math>
  
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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]].
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However, not every pair of matrices can be multiplied.  In order for matrices <math>A</math> and <math>B</math> to be compatible for multiplication, the number of ''columns'' in <math>A</math> must equal the number of ''rows'' in <math>B</math>.  If <math>A</math> is an <math>m \times n</math> matrix and <math>B</math> is an <math>n \times p</math> matrix, the product matrix <math>AB</math> will have <math>m</math> rows and <math>p</math> columns.
  
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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,
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Matrix multiplication is [[associativity|associative]]. However, matrix multiplication is ''not'' [[commutativity|commutative]]. That is, it is possible for <math>AB \neq BA</math>.
  
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<math>
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The <math>n \times n</math> [[identity matrix]] <math>I_n</math> satisfies the property:
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\sum_j(AB)_{ij}x_j = \sum_k A_{ik}(Bx)_k=\sum_{j,k}A_{ik}B_{kj}x_j
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<center><math>AI_n = I_n A = A</math></center>
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</math>
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for all <math>n \times n</math> matrices <math>A</math>.
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The vector <math>x</math> is arbitrary, so we can therefore deduce the rule for finding the product of two matrices:
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Moreover, every square matrix <math>A</math> with nonzero [[determinant]] has an inverse matrix <math>B</math> such that
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<center><math>AB = BA = I_n</math></center>
  
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<math>
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This means that for every positive integer <math>n</math>, the set of all <math>n \times n</math> matrices with nonzero determinant form a [[group]] under matrix multiplication.  This group is known as the [[general linear group]] <math>GL_{n}(\mathbb{R})</math>.
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(AB)_{ij}=\sum_k A_{ik}B_{kj}
 
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</math>
 
  
 
===Matrix concepts===
 
===Matrix concepts===
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*[[Basis]]
 
*[[Basis]]
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*[[Diagonalizable]]
 
*[[Eigenspace]]
 
*[[Eigenspace]]
 
*[[Eigenvalue]]
 
*[[Eigenvalue]]
 
*[[Eigenvector]]
 
*[[Eigenvector]]
 
*[[Gram-Schmidt process]]
 
*[[Gram-Schmidt process]]
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*[[Hermitian]]
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*[[Hermitian matrix]]
 
*[[Jordan canonical form]]
 
*[[Jordan canonical form]]
 
*[[Laplacian]]
 
*[[Laplacian]]
 
*[[Linear independence]]
 
*[[Linear independence]]
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*[[Matrix diagonalization]]
 
*[[Matrix reformation]]
 
*[[Matrix reformation]]
 
*[[Matrix transformation]]
 
*[[Matrix transformation]]
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*[[Resultant]]
 
*[[Resultant]]
 
*[[Span]]
 
*[[Span]]
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*[[Systems of linear equations]]
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*[[Linear equations]]
 
*[[Transcriptor]]
 
*[[Transcriptor]]
 
*[[Wronskian]]
 
*[[Wronskian]]
  
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[[Category:Linear algebra]]
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[[Category:Linear Algebra]]
 
[[Category:Computers]]
 
[[Category:Computers]]

Latest revision as of 12:23, September 17, 2017

For the 1999 film, see The Matrix.

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

To add two matrices, one would add their respective elements. For example:

<math>\begin{bmatrix} 

 x      & y & z      \\
 1 & 3 & 5 \\ 
 0      & 2 & 0

\end{bmatrix} + \begin{bmatrix} 

 0      & 3 & 1      \\
 4 & 3 & {x+2} \\ 
 0      & 4 & v

\end{bmatrix} </math>

would equal

<math>\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} </math>

Multiplication of matrices

To multiply two matrices, use the rule for finding the product of two matrices:

<math> (AB)_{ij}=\sum_k A_{ik}B_{kj} </math>

However, not every pair of matrices can be multiplied. In order for matrices <math>A</math> and <math>B</math> to be compatible for multiplication, the number of columns in <math>A</math> must equal the number of rows in <math>B</math>. If <math>A</math> is an <math>m \times n</math> matrix and <math>B</math> is an <math>n \times p</math> matrix, the product matrix <math>AB</math> will have <math>m</math> rows and <math>p</math> columns.

Matrix multiplication is associative. However, matrix multiplication is not commutative. That is, it is possible for <math>AB \neq BA</math>.

The <math>n \times n</math> identity matrix <math>I_n</math> satisfies the property:

<math>AI_n = I_n A = A</math>

for all <math>n \times n</math> matrices <math>A</math>.

Moreover, every square matrix <math>A</math> with nonzero determinant has an inverse matrix <math>B</math> such that

<math>AB = BA = I_n</math>

This means that for every positive integer <math>n</math>, the set of all <math>n \times n</math> matrices with nonzero determinant form a group under matrix multiplication. This group is known as the general linear group <math>GL_{n}(\mathbb{R})</math>.

Matrix concepts

Basic concepts

Advanced concepts

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