In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ' ...
, a matrix ring is a set of
matrices with entries in a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
''R'' that form a ring under
matrix addition
In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. However, there are other operations which could also be considered addition for matrices, such as the direct sum and the Krone ...
and
matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the ...
. The set of all matrices with entries in ''R'' is a matrix ring denoted M
''n''(''R'')
[Lang, ''Undergraduate algebra'', Springer, 2005; V.§3.] (alternative notations: Mat
''n''(''R'')
[ and ). Some sets of infinite matrices form infinite matrix rings. Any subring of a matrix ring is a matrix ring. Over a rng, one can form matrix rngs.
When ''R'' is a commutative ring, the matrix ring M''n''(''R'') is an ]associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplica ...
over ''R'', and may be called a matrix algebra. In this setting, if ''M'' is a matrix and ''r'' is in ''R'', then the matrix ''rM'' is the matrix ''M'' with each of its entries multiplied by ''r''.
Examples
* The set of all matrices over ''R'', denoted M''n''(''R''). This is sometimes called the "full ring of ''n''-by-''n'' matrices".
* The set of all upper triangular matrices over ''R''.
* The set of all lower triangular matrices over ''R''.
* The set of all diagonal matrices over ''R''. This subalgebra of M''n''(''R'') is isomorphic to the direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
of ''n'' copies of ''R''.
* For any index set ''I'', the ring of endomorphisms of the right ''R''-module is isomorphic to the ring of column finite matrices whose entries are indexed by and whose columns each contain only finitely many nonzero entries. The ring of endomorphisms of ''M'' considered as a left ''R''-module is isomorphic to the ring of row finite matrices.
* If ''R'' is a Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach sp ...
, then the condition of row or column finiteness in the previous point can be relaxed. With the norm in place, absolutely convergent series
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is sai ...
can be used instead of finite sums. For example, the matrices whose column sums are absolutely convergent sequences form a ring. Analogously of course, the matrices whose row sums are absolutely convergent series also form a ring. This idea can be used to represent operators on Hilbert spaces, for example.
* The intersection of the row finite and column finite matrix rings forms a ring .
*If ''R'' is commutative, then M''n''(''R'') has a structure of a *-algebra over ''R'', where the involution * on M''n''(''R'') is matrix transposition.
*If ''A'' is a C*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuou ...
, then Mn(''A'') is another C*-algebra. If ''A'' is non-unital, then Mn(''A'') is also non-unital. By the Gelfand-Naimark theorem, there exists a Hilbert space ''H'' and an isometric *-isomorphism from ''A'' to a norm-closed subalgebra of the algebra ''B''(''H'') of continuous operators; this identifies Mn(''A'') with a subalgebra of ''B''(''H''). For simplicity, if we further suppose that ''H'' is separable and ''A'' ''B''(''H'') is a unital C*-algebra, we can break up ''A'' into a matrix ring over a smaller C*-algebra. One can do so by fixing a projection ''p'' and hence its orthogonal projection 1 − ''p''; one can identify ''A'' with , where matrix multiplication works as intended because of the orthogonality of the projections. In order to identify ''A'' with a matrix ring over a C*-algebra, we require that ''p'' and 1 − ''p'' have the same ″rank″; more precisely, we need that ''p'' and 1 − ''p'' are Murray–von Neumann equivalent, i.e., there exists a partial isometry ''u'' such that ''p'' = ''uu''* and 1 − ''p'' = ''u''*''u''. One can easily generalize this to matrices of larger sizes.
* Complex matrix algebras M''n''(C) are, up to isomorphism, the only finite-dimensional simple associative algebras over the field C of complex numbers. Prior to the invention of matrix algebras, Hamilton Hamilton may refer to:
People
* Hamilton (name), a common British surname and occasional given name, usually of Scottish origin, including a list of persons with the surname
** The Duke of Hamilton, the premier peer of Scotland
** Lord Hamilto ...
in 1853 introduced a ring, whose elements he called biquaternions
In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions c ...
[Lecture VII of Sir William Rowan Hamilton, ''Lectures on quaternions'', Hodges and Smith, 1853.] and modern authors would call tensors in , that was later shown to be isomorphic to M''2''(C). One basis
Basis may refer to:
Finance and accounting
*Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
*Basis trading, a trading strategy consisting of ...
of M''2''(C) consists of the four matrix units (matrices with one 1 and all other entries 0); another basis is given by the identity matrix and the three Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
.
* A matrix ring over a field is a Frobenius algebra, with Frobenius form given by the trace of the product: .
Structure
* The matrix ring M''n''(''R'') can be identified with the ring of endomorphisms of the free right ''R''-module of rank ''n''; that is, . Matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the ...
corresponds to composition of endomorphisms.
* The ring M''n''(''D'') over a division ring ''D'' is an Artinian simple ring, a special type of semisimple ring. The rings and are ''not'' simple and not Artinian if the set ''I'' is infinite, but they are still full linear rings.
* The Artin–Wedderburn theorem states that every semisimple ring is isomorphic to a finite direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
, for some nonnegative integer ''r'', positive integers ''n''''i'', and division rings ''D''''i''.
*When we view Mn(C) as the ring of linear endomorphisms of Cn, those matrices which vanish on a given subspace V form a left ideal. Conversely, for a given left ideal ''I'' of Mn(C) the intersection of null spaces of all matrices in ''I'' gives a subspace of Cn. Under this construction, the left ideals of M''n''(C) are in bijection with the subspaces of Cn.
* There is a bijection between the two-sided ideals of M''n''(''R'') and the two-sided ideals of ''R''. Namely, for each ideal ''I'' of ''R'', the set of all matrices with entries in ''I'' is an ideal of M''n''(''R''), and each ideal of M''n''(''R'') arises in this way. This implies that M''n''(''R'') is simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
if and only if ''R'' is simple. For , not every left ideal or right ideal of M''n''(''R'') arises by the previous construction from a left ideal or a right ideal in ''R''. For example, the set of matrices whose columns with indices 2 through ''n'' are all zero forms a left ideal in M''n''(''R'').
* The previous ideal correspondence actually arises from the fact that the rings ''R'' and M''n''(''R'') are Morita equivalent. Roughly speaking, this means that the category of left ''R''-modules and the category of left M''n''(''R'')-modules are very similar. Because of this, there is a natural bijective correspondence between the ''isomorphism classes'' of left ''R''-modules and left M''n''(''R'')-modules, and between the isomorphism classes of left ideals of ''R'' and left ideals of M''n''(''R''). Identical statements hold for right modules and right ideals. Through Morita equivalence, M''n''(''R'') inherits any Morita-invariant properties of ''R'', such as being simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
, Artinian, Noetherian, prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
.
Properties
* If ''S'' is a subring of ''R'', then M''n''(''S'') is a subring of M''n''(''R''). For example, M''n''(Z) is a subring of M''n''(Q).
* The matrix ring M''n''(''R'') is commutative if and only if , , or ''R'' is commutative and . In fact, this is true also for the subring of upper triangular matrices. Here is an example showing two upper triangular matrices that do not commute, assuming :
*::
*:and
*::
* For ''n'' ≥ 2, the matrix ring M''n''(''R'') over a nonzero ring has zero divisors and nilpotent element
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the cla ...
s; the same holds for the ring of upper triangular matrices. An example in matrices would be
*::
* The center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentricity ...
of M''n''(''R'') consists of the scalar multiples of the identity matrix, , in which the scalar belongs to the center of ''R''.
* The unit group of M''n''(''R''), consisting of the invertible matrices under multiplication, is denoted GL''n''(''R'').
* If ''F'' is a field, then for any two matrices ''A'' and ''B'' in M''n''(''F''), the equality implies . This is not true for every ring ''R'' though. A ring ''R'' whose matrix rings all have the mentioned property is known as a stably finite ring .
Matrix semiring
In fact, ''R'' needs to be only a semiring for M''n''(''R'') to be defined. In this case, M''n''(''R'') is a semiring, called the matrix semiring. Similarly, if ''R'' is a commutative semiring, then M''n''(''R'') is a .
For example, if ''R'' is the Boolean semiring (the two-element Boolean algebra ''R'' = with 1 + 1 = 1),[Droste, M., & Kuich, W. (2009). Semirings and Formal Power Series. ''Handbook of Weighted Automata'', 3–28. ] then M''n''(''R'') is the semiring of binary relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and i ...
s on an ''n''-element set with union as addition, composition of relations
In the mathematics of binary relations, the composition of relations is the forming of a new binary relation from two given binary relations ''R'' and ''S''. In the calculus of relations, the composition of relations is called relative multiplica ...
as multiplication, the empty relation
In mathematics, a homogeneous relation (also called endorelation) over a set ''X'' is a binary relation over ''X'' and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on ''X''" or "a (binary) relation ...
(zero matrix In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero. It also serves as the additive identity of the additive group of m \times n matrices, and is denoted by the symbol O or 0 followed ...
) as the zero, and the identity relation ( identity matrix) as the unity.
See also
* Central simple algebra
In ring theory and related areas of mathematics a central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' which is simple, and for which the center is exactly ''K''. (Note that ''not'' every simple ...
* Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercom ...
* Hurwitz's theorem (normed division algebras)
* Generic matrix ring In algebra, a generic matrix ring is a sort of a universal matrix ring.
Definition
We denote by F_n a generic matrix ring of size ''n'' with variables X_1, \dots X_m. It is characterized by the universal property: given a commutative ring ''R'' a ...
* Sylvester's law of inertia
References
*
{{refend
Algebraic structures
Ring theory
Matrix theory