The < tt > MixColumns </ tt > step can also be viewed as a multiplication by a particular MDS matrix in a finite field.

* Any ring of matrices with coefficients in a commutative ring R forms an R-algebra under matrix addition and multiplication.

Other examples are readily found in different areas of mathematics, for example, vector addition, matrix multiplication and conjugation in groups.

For a finite-dimensional vector space, using a fixed orthonormal basis, the inner product can be written as a matrix multiplication of a row vector with a column vector:

and then it is understood that a bra next to a ket implies matrix multiplication.

The ket can be computed by normal matrix multiplication.

Then the bra can be computed by normal matrix multiplication.

For a finite-dimensional vector space, the outer product can be understood as simple matrix multiplication:

The advantage of doing this is that then all of the euclidean transformations become linear transformations and can be represented using matrix multiplication.

In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces ; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication.

The set of all 2 × 2 matrices is also a ring, under matrix addition and matrix multiplication.

* The group GL < sub > n </ sub >( R ) of invertible matrices ( under matrix multiplication ) is a Lie group of dimension n < sup > 2 </ sup >, called the general linear group.

While studying compositions of linear transformations, Arthur Cayley was led to define matrix multiplication and inverses.

* In matrix multiplication, there is actually a distinction between the cross and the dot symbols.

** matrix multiplication,

From the definition of matrix multiplication, there exists an × matrix, such that.

Cayley table of the symmetric group S < sub > 3 </ sub >( multiplication table of permutation matrix | permutation matrices ) These are the positions of the six matrices: File: Symmetric group 3 ; Cayley table ; positions. svg | 310px Only the unity matrices are arranged symmetrically to the main diagonal-thus the symmetric group is not abelian.

Ordinary vector algebra uses matrix multiplication to represent linear maps, and vector addition to represent translations.

Formally, in the finite-dimensional case, if the linear map is represented as a multiplication by a matrix A and the translation as the addition of a vector, an affine map acting on a vector can be represented as

Using an augmented matrix and an augmented vector, it is possible to represent both the translation and the linear map using a single matrix multiplication.

; Loop nest optimization: Some pervasive algorithms such as matrix multiplication have very poor cache behavior and excessive memory accesses.

The sum and product of dual numbers are then calculated with ordinary matrix addition and matrix multiplication ; both operations are commutative and associative.

The determinant of a matrix A is denoted det ( A ), det A, or | A |.

Logical matrix | Logical matrices of the Bell number | 52 equivalence relations on a 5-element set ( Colored fields, including those in light gray, stand for ones ; white fields for zeros.

; Incidence matrix: The graph is represented by a matrix of size | V | ( number of vertices ) by | E | ( number of edges ) where the entry edge contains the edge's endpoint data ( simplest case: 1-incident, 0-not incident ).

A Tandy 1000 HX, with a Tandy RGB monitor, an external 5. 25 disk drive, joystick, and a Tandy DMP-133 dot matrix Computer printer | printer.

Upper image: Inmac ink ribbon Cartridge ( electronics ) | cartridge with black ink for Dot matrix printerLower image: Inked and folded, the ribbon is pulled into the cartridge by the roller mechanism to the left.

The column vectors of a matrix ( mathematics ) | matrix.

File: Malachite-Copper-264040. jpg | Green clusters of malachite crystals fill the vugs in a matrix of solid, hackly native copper.

Visualization of the SVD of a two-dimensional, real Shear mapping | shearing matrix M. First, we see the unit disc in blue together with the two standard basis | canonical unit vectors.

The SVD decomposes M into three simple transformations: a Rotation matrix | rotation V < sup >*</ sup >, a Scaling matrix | scaling Σ along the rotated coordinate axes and a second rotation U. The lengths σ < sub > 1 </ sub > and σ < sub > 2 </ sub > of the Ellipse # Elements of an ellipse | semi-axes of the ellipse are the singular value s of M.

The outer product is an N × N matrix, as expected for a linear operator.

Equivalently, the determinant can be expressed as a sum of products of entries of the matrix where each product has n terms and the coefficient of each product is − 1 or 1 or 0 according to a given rule: it is a polynomial expression of the matrix entries.

The first part of the algorithm computes an LU decomposition, while the second part writes the original matrix as the product of a uniquely determined invertible matrix and a uniquely determined reduced row-echelon matrix.

As explained above, Gaussian elimination writes a given m × n matrix A uniquely as a product of an invertible m × m matrix S and a row-echelon matrix T. Here, S is the product of the matrices corresponding to the row operations performed.

Remark: Some authors, especially in physics and matrix algebra, prefer to define the inner product and the sesquilinear form with linearity in the second argument rather than the first.

In those disciplines we would write the product as ( the bra-ket notation of quantum mechanics ), respectively ( dot product as a case of the convention of forming the matrix product AB as the dot products of rows of A with columns of B ).

When light crosses an optical element the resulting polarization of the emerging light is found by taking the product of the Jones matrix of the optical element and the Jones vector of the incident light.

Therefore, the spin angular velocity vector () about the spin axis will have to evolve in time so that the matrix product remains constant.

Also not surprisingly, the density matrix of for the pure product state

Here a rank 1 tensor ( matrix product of a column vector and a row vector ) is the same thing as a rank 1 matrix of the given size.