Difference between revisions of "Matrix"

Revision as of 19:42, October 15, 2009

Did you take the red pill or the blue pill?

@@ Line 1: / Line 1: @@
-''For the 1999 film, see [[The Matrix]].''
+Did you take the red pill or the blue pill?
-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.
-==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:
-<math>\begin{bmatrix}
-  x      & y & z      \\
-& 3 & 5 \\
-      & 2 & 0
-\end{bmatrix} + \begin{bmatrix}
-      & 3 & 1      \\
-& 3 & {x+2} \\
-      & 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} \\
-& 6 & {x+7} \\
-& 6 & v
-\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>
-===Matrix concepts===
-====Basic concepts====
-*[[Adjoint]]
-*[[Determinant]]
-*[[Diagonal matrix]]
-*[[Identity matrix]]
-*[[Inverse matrix]]
-*[[Null, column and row space]]
-*[[Scalar]]
-*[[Trace]]
-*[[Transpose matrix]]
-*[[Vector]]
-*[[Zero matrix]]
-====Advanced concepts====
-*[[Basis]]
-*[[Diagonalizeable]]
-*[[Eigenspace]]
-*[[Eigenvalue]]
-*[[Eigenvector]]
-*[[Gram-Schmidt process]]
-*[[Hermitian matrix]]
-*[[Jordan canonical form]]
-*[[Laplacian]]
-*[[Linear independence]]
-*[[Matrix diagonalization]]
-*[[Matrix reformation]]
-*[[Matrix transformation]]
-*[[Orthogonal matrix]]
-*[[Orthonormal matrix]]
-*[[Resultant]]
-*[[Span]]
-*[[Systems of linear equations]]
-*[[Transcriptor]]
-*[[Wronskian]]
-[[Category:Linear algebra]]
-[[Category:Computers]]

Difference between revisions of "Matrix"

Revision as of 19:42, October 15, 2009

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