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.

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:

would equal

Multiplication of matrices

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

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