Matrix - Revision history

FredericBernard: /* Advanced concepts */ Capatalise

2017-09-17T12:23:29Z

Advanced concepts: Capatalise

FredericBernard: Fixed broken link

2017-09-11T16:01:43Z

Fixed broken link

DavidB4-bot: /* Advanced concepts */Category

2016-07-28T14:35:21Z

Advanced concepts: Category

DavidB4-bot: /* top */clean up

2016-06-23T15:00:37Z

top: clean up

GregG: /* Multiplication of matrices */ expand

2012-07-01T00:53:19Z

Multiplication of matrices: expand

GregG: /* Multiplication of matrices */ rewrite this section

2012-07-01T00:49:01Z

Multiplication of matrices: rewrite this section

GregG: /* Addition of matrices */ rewrite this sentence

2012-07-01T00:44:42Z

Addition of matrices: rewrite this sentence

DavidE: /* Multiplication of matrices */ internal link

2012-02-07T18:18:22Z

Multiplication of matrices: internal link

Ed Poor: Matrix algebra

2010-02-13T20:09:55Z

Matrix algebra

DanieleGiusto: /* Advanced concepts */

2010-01-07T21:46:33Z

Advanced concepts

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 *[[Resultant]]
 *[[Span]]
-*[[linear equations]]
+*[[Linear equations]]
 *[[Transcriptor]]
 *[[Wronskian]]

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 *[[Resultant]]
 *[[Span]]
-*[[Systems of linear equations]]
+*[[linear equations]]
 *[[Transcriptor]]
 *[[Wronskian]]

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 *[[Wronskian]]
-[[Category:Linear algebra]]
+[[Category:Linear Algebra]]
 [[Category:Computers]]

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 ''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]].

← Older revision		Revision as of 00:53, July 1, 2012
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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>.		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===		===Matrix concepts===

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 ===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:
+To multiply two matrices, use the rule for finding the product of two matrices:
-<math>\begin{bmatrix}
+<math>
-a & b \\
+(AB)_{ij}=\sum_k A_{ik}B_{kj}
 </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]].
+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.
-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,
+Matrix multiplication is [[associativity|associative]].  However, matrix multiplication is ''not'' [[commutativity|commutative]].  That is, it is possible for <math>AB \neq BA</math>.
 ===Matrix concepts===

@@ Line 11: / Line 11: @@
 ===Addition of matrices===
-For example, to add two matrices, one would add their respective elements, thus:
+To add two matrices, one would add their respective elements.  For example:
 <math>\begin{bmatrix}

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 *[[Basis]]
-*[[Diagonalizeable]]
+*[[Diagonalizable]]
 *[[Eigenspace]]
 *[[Eigenvalue]]