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:

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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:

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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 $B$. If $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix, the product matrix $AB$ will have $m$ rows and $p$ columns.

Matrix multiplication is associative. However, matrix multiplication is not commutative. That is, it is possible for $AB\neq BA$.

The $n\times n$ identity matrix $I_{n}$ satisfies the property:

$AI_{n}=I_{n}A=A$

for all $n\times n$ matrices $A$.

Moreover, every square matrix $A$ 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

• 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
• linear equations
• Transcriptor
• Wronskian