Difference between revisions of "Matrix"

Revision as of 00:53, July 1, 2012

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:

${\begin{bmatrix}x&y&z\\1&3&5\\0&2&0\end{bmatrix}}+{\begin{bmatrix}0&3&1\\4&3&{x+2}\\0&4&v\end{bmatrix}}$

would equal

${\begin{bmatrix}{x+0}&{y+3}&{z+1}\\{1+4}&{3+3}&{5+(x+2)}\\{0+0}&{2+4}&{0+v}\end{bmatrix}}={\begin{bmatrix}x&{y+3}&{z+1}\\5&6&{x+7}\\0&6&v\end{bmatrix}}$

Multiplication of matrices

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

$(AB)_{ij}=\sum _{k}A_{ik}B_{kj}$

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

Matrix multiplication is associative. However, matrix multiplication is not commutative. That is, it is possible for $AB\neq BA$ .

The $n\times n$ identity matrix $I_{n}$ satisfies the property:

AI_{n}=I_{n}A=A

for all $n\times n$ matrices $A$ .

Moreover, every square matrix 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} with nonzero determinant has an inverse matrix 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} such that

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 = BA = I_n}

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

Matrix concepts

Basic concepts

Advanced concepts

@@ Line 45: / Line 45: @@
 Matrix multiplication is [[associativity|associative]].  However, matrix multiplication is ''not'' [[commutativity|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:
+<center><math>AI_n = I_n A = A</math></center>
+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
+<center><math>AB = BA = I_n</math></center>
+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===