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.

## Contents

* 1 Mathematics
* 1.1 Addition of matrices
* 1.2 Multiplication of matrices
* 1.3 Matrix concepts
* 1.3.1 Basic concepts
* 1.3.2 Advanced concepts

## Mathematics[edit]

In mathematics, matrices can be manipulated in a variety of ways, including addition and multiplication.

### Addition of matrices[edit]

To add two matrices, one would add their respective elements. For example:

would equal

### Multiplication of matrices[edit]

To multiply two matrices, use the rule for finding the product of two matrices:

However, not every pair of matrices can be multiplied. In order for matrices  and  to be compatible for multiplication, the number of columns in  must equal the number of rows in . If  is an  matrix and  is an  matrix, the product matrix  will have  rows and  columns.

Matrix multiplication is associative. However, matrix multiplication is not commutative. That is, it is possible for .

The  identity matrix satisfies the property:

for all  matrices .

Moreover, every square matrix  with nonzero determinant has an inverse matrix  such that

This means that for every positive integer , the set of all  matrices with nonzero determinant form a group under matrix multiplication. This group is known as the general linear group .

### Matrix concepts[edit]

#### Basic concepts[edit]

* Adjoint
* Determinant
* Diagonal matrix
* Identity matrix
* Inverse matrix
* Null, column and row space
* Scalar
* Trace
* Transpose matrix
* Vector
* Zero matrix

#### Advanced concepts[edit]

* 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