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| {{Redirect|Matrix algebra|the algebraic theory of matrices|Matrix (mathematics)|and|Linear algebra}}
| | The name of creator is Roberto but he never really liked that name. One of things she loves most is caving and he or she would never give it up. New Jersey is the place Really like most nevertheless will need to move within a year or two. My day job is a standard control and order filler but I've already tried for another one particular particular. He's been functioning on his website for some time now. Certify it out here: http://m.scalarenergypendants.com/<br><br>My blog [http://m.scalarenergypendants.com/ Quantum Science Pendants] |
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| In [[abstract algebra]], a '''matrix ring''' is any collection of [[matrix (mathematics)|matrices]] forming a [[ring (mathematics)|ring]] under [[matrix addition]] and [[matrix multiplication]]. The set of ''n''×''n'' matrices with entries from another ring is a matrix ring, as well as some subsets of infinite matrices which form '''infinite matrix rings'''. Any subrings of these matrix rings are also called matrix rings.
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| In the case when ''R'' is a commutative ring, then the matrix ring M<sub>''n''</sub>(''R'') is an [[associative algebra]] which may be called a '''matrix algebra'''. In this situation, if ''M'' is a matrix and ''r'' is in ''R'', then the matrix ''Mr'' is the matrix ''M'' with each of its entries multiplied by ''r''.
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| It is assumed throughout that ''R'' is an [[associative ring]] with a unit 1 ≠ 0, although matrix rings can be formed over rings without unity.
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| == Examples ==
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| * The set of all ''n''×''n'' matrices over an arbitrary ring ''R'', denoted M<sub>''n''</sub>(''R''). This is usually referred to as the "full ring of ''n'' by ''n'' matrices". These matrices represent endomorphisms of the free module ''R''<sup>''n''</sup>.
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| * The set of all upper (or set of all lower) [[triangular matrices]] over a ring.
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| * If ''R'' is any ring with unity, then the ring of endomorphisms of <math>M=\bigoplus_{i\in I}R</math> as a right ''R'' module is isomorphic to the ring of '''column finite matrices''' <math>\mathbb{CFM}_I(R)\,</math> whose entries are indexed by <math>I\times I</math>, and whose columns each contain only finitely many nonzero entries. The endomorphisms of ''M'' considered as a left ''R'' module result in an analogous object, the '''row finite matrices''' <math>\mathbb{RFM}_I(R)</math> whose rows each only have finitely many nonzero entries.
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| * If ''R'' is a [[normed algebra|normed ring]], 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 [[Hilbert space#Operators on Hilbert spaces|operators on Hilbert spaces]], for example.
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| * The intersection of the row and column finite matrix rings also forms a ring, which can be denoted by <math>\mathbb{RCFM}_I(R)\,</math>.
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| * The algebra M<sub>2</sub>('''R''') of [[2 × 2 real matrices]] is a simple example of a non-commutative associative algebra. Like the [[quaternion]]s, it has [[dimension (linear algebra)|dimension]] 4 over '''R''', but unlike the quaternions, it has [[zero divisor]]s, as can be seen from the following product of the [[matrix unit]]s: ''E''<sub>11</sub>''E''<sub>21</sub> = 0, hence it is not a [[division ring]]. Its invertible elements are [[nonsingular matrix|nonsingular matrices]] and they form a [[group (mathematics)|group]], the [[general linear group]] ''GL''(2,'''R''').
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| *If ''R'' is [[commutative ring|commutative]], the matrix ring has a structure of a [[*-algebra]] over ''R'', where the [[involution (mathematics)#Ring theory|involution]] * on M<sub>''n''</sub>(''R'') is the [[matrix transpose|matrix transposition]].
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| * Complex matrix algebras M<sub>''n''</sub>('''C''') are, up to isomorphism, the only simple associative algebras over the field '''C''' of [[complex number]]s. For ''n'' = 2, the matrix algebra M<sub>''2''</sub>('''C''') plays an important role in the theory of [[angular momentum]]. It has an alternative basis given by the [[identity matrix]] and the three [[Pauli matrices]]. M<sub>''2''</sub>('''C''') was the scene of early abstract algebra in the form of [[biquaternion]]s.
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| * A matrix ring over a field is a [[Frobenius algebra]], with Frobenius form given by the trace of the product: ''σ''(''A'',''B'')=tr(''AB'').
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| ==Structure==
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| * The matrix ring M<sub>n</sub>(''R'') can be identified with the [[ring of endomorphisms]] of the [[free module|free ''R''-module]] of rank ''n'', M<sub>''n''</sub>(''R'') ≅ End<sub>''R''</sub>(''R''<sup>''n''</sup>). The procedure for [[matrix multiplication]] can be traced back to compositions of endomorphisms in this endomorphism ring.
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| * The ring M<sub>n</sub>(''D'') over a [[division ring]] ''D'' is an [[Artinian ring|Artinian]] [[simple ring]], a special type of [[semisimple ring]]. The rings <math>\mathbb{CFM}_I(D)</math> and <math>\mathbb{RFM}_I(D)</math> are ''not'' simple and not Artinian if the set <math>I\,</math> is infinite, however they are still [[full linear ring]]s.
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| * In general, every semisimple ring is isomorphic to a finite direct product of full matrix rings over division rings, which may have differing division rings and differing sizes. This classification is given by the [[Artin–Wedderburn theorem]].
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| * There is a one-to-one correspondence between the two-sided [[ideal (ring theory)|ideals]] of M<sub>''n''</sub>(''R'') and the two-sided ideals of ''R''. Namely, for each ideal ''I'' of ''R'', the set of all ''n''×''n'' matrices with entries in ''I'' is an ideal of M<sub>''n''</sub>(''R''), and each ideal of M<sub>''n''</sub>(''R'') arises in this way. This implies that M<sub>''n''</sub>(''R'') is [[simple ring|simple]] if and only if ''R'' is simple. For ''n'' ≥ 2, not every left ideal or right ideal of M<sub>''n''</sub>(''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<sub>''n''</sub>(''R'').
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| * The previous ideal correspondence actually arises from the fact that the rings ''R'' and M<sub>''n''</sub>(''R'') are [[Morita equivalent]]. Roughly speaking, this means that the category of left ''R'' modules and the category of left M<sub>''n''</sub>(''R'') modules are very similar. Because of this, there is a natural bijective correspondence between the ''isomorphism classes'' of the left ''R''-modules and the left M<sub>''n''</sub>(''R'')-modules, and between the isomorphism classes of the left ideals of ''R'' and M<sub>''n''</sub>(''R''). Identical statements hold for right modules and right ideals. Through Morita equivalence, M<sub>''n''</sub>(''R'') can inherit any properties of ''R'' which are Morita invariant, such as being [[simple ring|simple]], [[artinian ring|Artinian]], [[noetherian ring|Noetherian]], [[prime ring|prime]] and numerous other properties as given in the [[Morita equivalence]] article.
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| == Properties ==
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| * The matrix ring M<sub>''n''</sub>(''R'') is [[commutative ring|commutative]] if and only if ''n'' = 1 and ''R'' is [[commutative ring|commutative]]. In fact, this is also true for the subring of upper triangular matrices. Here is an example for 2×2 matrices (in fact, upper triangular matrices) which do not commute:
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| :<math> | |
| \begin{bmatrix}
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| 1 & 0 \\
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| 0 & 0
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| \end{bmatrix}
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| \begin{bmatrix}
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| 1 & 1 \\
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| 0 & 0
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| 1 & 1 \\
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| 0 & 0
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| \end{bmatrix}\,
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| </math>
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| and
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| <math> \begin{bmatrix}
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| 1 & 1 \\
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| 0 & 0
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| \end{bmatrix}
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| \begin{bmatrix}
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| 1 & 0 \\
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| 0 & 0
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| 1 & 0 \\
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| 0 & 0
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| \end{bmatrix}\,
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| </math>. This example is easily generalized to ''n''×''n'' matrices.
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| * For ''n'' ≥ 2, the matrix ring M<sub>''n''</sub>(''R'') has [[zero divisor]]s and [[nilpotent element]]s, and again, the same thing can be said for the upper triangular matrices. An example in 2×2 matrices would be
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| :<math> \begin{bmatrix} | |
| 0 & 1 \\
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| 0 & 0
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| \end{bmatrix}
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| \begin{bmatrix}
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| 0 & 1 \\
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| 0 & 0
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| 0 & 0 \\
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| 0 & 0
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| \end{bmatrix}\,
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| </math>.
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| * The [[center (algebra)|center]] of a matrix ring over a ring ''R'' consists of the matrices which are scalar multiples of the [[identity matrix]], where the scalar belongs to the [[center (algebra)|center]] of ''R''.
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| * In linear algebra, it is noted that over a field ''F'', M<sub>''n''</sub>(''F'') has the property that for any two matrices ''A'' and ''B'', ''AB''=1 implies ''BA''=1. 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''' or sometimes '''weakly finite ring''' {{harv|Lam|1999|p=5}}.
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| == See also ==
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| * [[Central simple algebra]]
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| * [[Clifford algebra]]
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| * [[Hurwitz's theorem (normed division algebras)]]
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| ==References==
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| * {{Citation | last1=Lam | first1=T. Y. | title=Lectures on modules and rings | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics No. 189 | isbn=978-0-387-98428-5 | year=1999}}
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| [[Category:Algebraic structures]]
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| [[Category:Ring theory]]
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| [[Category:Matrix theory]]
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The name of creator is Roberto but he never really liked that name. One of things she loves most is caving and he or she would never give it up. New Jersey is the place Really like most nevertheless will need to move within a year or two. My day job is a standard control and order filler but I've already tried for another one particular particular. He's been functioning on his website for some time now. Certify it out here: http://m.scalarenergypendants.com/
My blog Quantum Science Pendants