For a Circulant matrix, the singular value decomposition is given by the Fourier transform,

* Circulant matrix, a Toeplitz matrix with the additional property that.

* Gray R. M., Toeplitz and Circulant Matrices: A Review ( Now Publishers ).

* The square n-by-n matrices with entries from the field K form a unitary associative algebra over K.

* The 2 × 2 real matrices form an associative algebra useful in plane mapping.

If the 3 generations are then put in a 3x3 hermitian matrix with certain additions for the diagonal elements then these matrices form an exceptional ( grassman -) Jordan algebra, which has the symmetry group of one of the exceptional Lie groups ( F < sub > 4 </ sub >, E < sub > 6 </ sub >, E < sub > 7 </ sub > or E < sub > 8 </ sub >) depending on the details.

Stated equivalently for matrices, the first part reduces a matrix to row echelon form using elementary row operations while the second reduces it to reduced row echelon form, or row canonical form.

In this way, for example, some matrices can be transformed to a matrix that has a reduced row echelon form like

* The 2 × 2 real invertible matrices form a group under multiplication, denoted by GL < sub > 2 </ sub >( R ):

* The rotation matrices form a subgroup of GL < sub > 2 </ sub >( R ), denoted by SO < sub > 2 </ sub >( R ).

Unitary matrices of determinant 1 form a closed connected subgroup of dimension n < sup > 2 </ sup > − 1 denoted SU ( n ), the special unitary group.

* The symplectic group Sp < sub > 2n </ sub >( R ) consists of all 2n × 2n matrices preserving a symplectic form on R < sup > 2n </ sup >.

The Pauli matrices ( after multiplication by i to make them anti-hermitian ), also generate transformations in the sense of Lie algebras: the matrices form a basis for, which exponentiates to the spin group, and for the identical Lie algebra, which exponentiates to the Lie group of rotations of 3-dimensional space.

* Together with the identity matrix I ( which is sometimes written as σ < sub > 0 </ sub >), the Pauli matrices form an orthogonal basis, in the sense of Hilbert-Schmidt, for the real Hilbert space of 2 × 2 complex Hermitian matrices, or the complex Hilbert space of all 2 × 2 matrices.

A potentially easier way to identify a matrices ' rank is to use elementary row operations to put the matrix in reduced row-echelon form and simply count the number of non-zero rows in the matrix.

These matrices, and the form of the wave function, have a deep mathematical significance.

Although the output of modern computers is generally all in the form of dot matrices, computers may internally store data as either a dot matrix or as a vector pattern of lines and curves.

However, matrices that can be diagonalized with the same similarity transformation do form a commutative ring.

( In post-Crisis on Infinite Earths continuity, the Eradicator, an artificial life form programmed to preserve all Kryptonian culture, altered the birthing matrices (" artificial wombs ") that the explorers took with them so that all newborns would be fatally vulnerable to lead and other materials such as greenhouse gases and certain rocks.

A matrix can also be factorized into a product of matrices of special types, for an application in which that form is convenient.

The researchers also found TEIQue scores were unrelated to nonverbal reasoning ( Raven's matrices ), which they interpreted as support for the personality trait view of EI ( as opposed to a form of intelligence ).

Most kinds of numbers ( example 1 ) and matrices ( example 4 ) form rings.

The n × n orthogonal matrices form a group under matrix multiplication, the orthogonal group denoted by O ( n ), which — with its subgroups — is widely used in mathematics and the physical sciences.

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

Although most often used for matrices whose entries are real or complex numbers, the definition of the determinant only involves addition, subtraction and multiplication, and so it can be defined for square matrices with entries taken from any commutative ring.

For square matrices with entries in a non-commutative ring, for instance the quaternions, there is no unique definition for the determinant, and no definition that has all the usual properties of determinants over commutative rings.

The ring of 2 × 2 matrices is not commutative, since matrix multiplication fails to be commutative, as the following example shows:

Over a commutative ring R, one must be slightly more careful: a matrix over R is invertible if and only if its determinant is a unit in R, that is, if its determinant is invertible in R. Therefore GL ( n, R ) may be defined as the group of matrices whose determinants are units.

However, for matrices over a more general ring that need not be commutative, such as the quaternions, they may not be equal.

This identity holds for any matrices over a commutative ring, but not for all rings in general .</ li >

The determinant of a product AB is the product of the determinants of matrices A and B ( not defined when the underlying ring is not commutative ):

The fact that commuting matrices have a common eigenvector – and hence by induction stabilize a common flag and are simultaneously triangularizable – can be interpreted as a result of the weak Nullstellensatz, as follows: commuting matrices form a commutative algebra

We shall therefore now consider only arguments that prove the theorem directly for any matrix using algebraic manipulations only ; these also have the benefit of working for matrices with entries in any commutative ring.

Being a consequence of just algebraic expression manipulation, these relations are valid for matrices with entries in any commutative ring ( commutativity must be assumed for determinants to be defined in the first place ).

* the algebra of all n-by-n matrices over the field ( or commutative ring ) K. Here the multiplication is ordinary matrix multiplication.

Because addition of vectors is commutative, multiplication of translation matrices is therefore also commutative ( unlike multiplication of arbitrary matrices ).

The determinant is over the ring of n × n matrices whose entries are polynomials in t with coefficients in the commutative algebra of even complex differential forms on M. The curvature form of V is defined as

The formula is valid for matrices with entries from any commutative ring.

More generally, this likewise applies to the square matrices whose entries are elements of any other given semiring S, and the semiring is generally non-commutative even though S may be commutative.

Abstract linear algebra considers matrices with entries from any commutative ring, not limited to the integers.

More formally, the group algebra of the Heisenberg group has center so rather than simply thinking of the group algebra as an algebra over the field of scalars K, one may think of it as an algebra over the commutative algebra As the center of a matrix algebra or operator algebra is the scalar matrices, a-structure on the matrix algebra is a choice of scalar matrix – a choice of scale.