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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. − − 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 \\ − 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, 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]]
Replaced content with 'Did you take the red pill or the blue pill?'
Did you take the red pill or the blue pill?