In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
, a matrix ring is a set of
matrices
Matrix (: matrices or matrixes) or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics), a rectangular array of numbers, symbols or expressions
* Matrix (logic), part of a formula in prenex normal form
* Matrix (biology), the ...
with entries in a
ring ''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.
For a vector, \vec\!, adding two matrices would have the geometric effect of applying each matrix transformation separately ...
and
matrix multiplication
In mathematics, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the n ...
. The set of all matrices with entries in ''R'' is a matrix ring denoted M
''n''(''R'') (alternative notations: Mat
''n''(''R'') and ). Some sets of infinite matrices form infinite matrix rings. A subring of a matrix ring is again 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'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a mult ...
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
square 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
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagona ...
over ''R''. This
subalgebra In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations.
"Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear opera ...
of M
''n''(''R'') is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to the
direct product 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, then the condition of row or column finiteness in the previous point can be relaxed. With the norm in place,
absolutely convergent series 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
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
, then M
''n''(''R'') has a structure of a
*-algebra over ''R'', where the
involution
Involution may refer to: Mathematics
* Involution (mathematics), a function that is its own inverse
* Involution algebra, a *-algebra: a type of algebraic structure
* Involute, a construction in the differential geometry of curves
* Exponentiati ...
* 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 contin ...
, then M
''n''(''A'') is another C*-algebra. If ''A'' is non-unital, then M
''n''(''A'') is also non-unital. By the
Gelfand–Naimark theorem, there exists a
Hilbert space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
''H'' and an isometric *-isomorphism from ''A'' to a norm-closed subalgebra of the algebra ''B''(''H'') of continuous operators; this identifies M
''n''(''A'') with a subalgebra of ''B''(''H''
⊕''n''). 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
Projection or projections may refer to:
Physics
* Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction
* The display of images by a projector
Optics, graphics, and carto ...
''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
Partial may refer to:
Mathematics
*Partial derivative, derivative with respect to one of several variables of a function, with the other variables held constant
** ∂, a symbol that can denote a partial derivative, sometimes pronounced "partial ...
''u'' such that and . 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 number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s. Prior to the invention of matrix algebras,
Hamilton
Hamilton may refer to:
* Alexander Hamilton (1755/1757–1804), first U.S. Secretary of the Treasury and one of the Founding Fathers of the United States
* ''Hamilton'' (musical), a 2015 Broadway musical by Lin-Manuel Miranda
** ''Hamilton'' (al ...
in 1853 introduced a ring, whose elements he called
biquaternions[Lecture VII of Sir William Rowan Hamilton (1853) ''Lectures on Quaternions'', Hodges and Smith] and modern authors would call tensors in , that was later shown to be isomorphic to M
2(C). One
basis 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
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the obje ...
and the three
Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () ...
.
* 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, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the n ...
corresponds to composition of endomorphisms.
* The ring M
''n''(''D'') over a
division ring
In algebra, a division ring, also called a skew field (or, occasionally, a sfield), is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicativ ...
''D'' is an
Artinian simple ring In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field.
The center of a sim ...
, a special type of
semisimple ring
In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itself ...
. The rings
and
are ''not'' simple and not Artinian if the set ''I'' is infinite, but they are still
full linear ring
In the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic ...
s.
* The
Artin–Wedderburn theorem states that every semisimple ring is isomorphic to a finite
direct product , for some nonnegative integer ''r'', positive integers ''n''
''i'', and division rings ''D''
''i''.
* When we view M
''n''(C) as the ring of linear endomorphisms of C
''n'', those matrices which vanish on a given subspace ''V'' form a
left ideal
In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even n ...
. Conversely, for a given left ideal ''I'' of M
''n''(C) the intersection of
null spaces of all matrices in ''I'' gives a subspace of C
''n''. Under this construction, the left ideals of M
''n''(C) are in bijection with the subspaces of C
''n''.
* 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 John ...
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 John ...
,
Artinian,
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite leng ...
,
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
In mathematics, a subring of a ring is a subset of that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and that shares the same multiplicative identity as .In general, not all s ...
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
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
if and only if , , or ''R'' is
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
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 in ''R'':
*::
*: and
*::
* For , the matrix ring M
''n''(''R'') over a
nonzero ring has
zero divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right ze ...
s 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, along with its sister idempotent, was introduced by Benjamin Peirce i ...
s; the same holds for the ring of upper triangular matrices. An example in matrices would be
*::
* The
center of M
''n''(''R'') consists of the scalar multiples of the
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the obje ...
, ''I''
''n'', in which the scalar belongs to the center of ''R''.
* The
unit group
In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that
vu = uv = 1,
where is the multiplicative identity; the el ...
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
In abstract algebra, a semiring is an algebraic structure. Semirings are a generalization of rings, dropping the requirement that each element must have an additive inverse. At the same time, semirings are a generalization of bounded distribu ...
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
Any kind of logic, function, expression, or theory based on the work of George Boole is considered Boolean.
Related to this, "Boolean" may refer to:
* Boolean data type, a form of data with only two possible values (usually "true" and "false" ...
(the
two-element Boolean algebra
In mathematics and abstract algebra, the two-element Boolean algebra is the Boolean algebra whose ''underlying set'' (or universe or ''carrier'') ''B'' is the Boolean domain. The elements of the Boolean domain are 1 and 0 by convention, so that ''B ...
with ), then M
''n''(''R'') is the semiring of
binary relation
In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs ...
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 (
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 followe ...
) as the zero, and the
identity relation
In mathematics, a homogeneous relation (also called endorelation) on a set ''X'' is a binary relation between ''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 ...
(
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the obje ...
) 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'' that 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 with the additional structure of a distinguished subspace. As -algebras, they generalize the real number ...
*
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'' ...
*
Sylvester's law of inertia
Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of basis. Namely, if A is a symmetric matrix, then for any invertible matr ...
Citations
References
*
*
*
*
*
*
* , corrected 5th printing
{{refend
Algebraic structures
Ring theory
Matrix theory