Difference between revisions of "Matrix"

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:

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

would equal

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, use 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} }

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

Matrix multiplication is associative. However, matrix multiplication is not commutative. That is, it is possible for 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} .

The 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} identity 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 I_n} satisfies the property:

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 AI_n = I_n A = A}

for 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 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} .

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 1: / Line 1: @@
 ''For the 1999 film, see [[The Matrix]].''
-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]]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]].
@@ 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===
@@ Line 81: / Line 90: @@
 *[[Resultant]]
 *[[Span]]
-*[[Systems of linear equations]]
+*[[Linear equations]]
 *[[Transcriptor]]
 *[[Wronskian]]
-[[Category:Linear algebra]]
+[[Category:Linear Algebra]]
 [[Category:Computers]]