Matrix

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:

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 & 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, 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:

${\begin{bmatrix}a&b\\c&d\end{bmatrix}}{\begin{bmatrix}e\\f\end{bmatrix}}={\begin{bmatrix}ae+bf\\ce+df\end{bmatrix}}$

It is important to note that matrix multiplication is not commutative: in general, $AB\neq BA$ for two matrices $A$ and $B$. 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, $AB$, which is itself a matrix. Then form the product $ABx$. Matrix multiplication is associative, so we can consider this as either $(AB)x$ or $A(Bx)$. In suffix notation,

$\sum _{j}(AB)_{ij}x_{j}=\sum _{k}A_{ik}(Bx)_{k}=\sum _{j,k}A_{ik}B_{kj}x_{j}$ The vector 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 x} is arbitrary, so we can therefore deduce 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} }

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
• Diagonalizable
• 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